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Easy2Hard-Bench

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HMMT

HMMT-Feb

guts

Feb

Let $A B C$ be an equilateral triangle with side length 1. Points $D, E, F$ lie inside triangle $A B C$ such that $A, E, F$ are collinear, $B, F, D$ are collinear, $C, D, E$ are collinear, and triangle $D E F$ is equilateral. Suppose that there exists a unique equilateral triangle $X Y Z$ with $X$ on side $\overline{B C}, Y$ on side $\overline{A B}$, and $Z$ on side $\overline{A C}$ such that $D$ lies on side $\overline{X Z}, E$ lies on side $\overline{Y Z}$, and $F$ lies on side $\overline{X Y}$. Compute $A Z$.

\frac{1}{1+\sqrt[3]{2}}

\section*{Solution:} First, note that point $X$ can be constructed from intersection of $\odot(D O F)$ and side $\overline{B C}$. Thus, if there is a unique equilateral triangle, then we must have that $\odot(D O F)$ is tangent to $\overline{B C}$. Furthermore, $\odot(D O F)$ is tangent to $D E$, so by equal tangents, we have $C D=C X$. We now compute the answer. Let $x=A Z=C X=C D=B F$. Then, by power of point, \[ B F \cdot B D=B X^{2} \Longrightarrow B D=\frac{(1-x)^{2}}{x} \] Thus, by law of cosine on $\triangle B D C$, we have that \[ \begin{aligned} x^{2}+\left(\frac{(1-x)^{2}}{x}\right)^{2}+x \cdot \frac{(1-x)^{2}}{x} & =1 \\ x^{2}+\frac{(1-x)^{4}}{x^{2}}+(1-x)^{2} & =1 \\ \frac{(1-x)^{4}}{x^{2}} & =2 x(1-x) \\ \frac{1-x}{x} & =\sqrt[3]{2} \\ x & =\frac{1}{1+\sqrt[3]{2}} \end{aligned} \] $\fbox{\frac{1}{1+\sqrt[3]{2}}}$.

HMMT Feb Guts

HMMT-Feb Guts

AMC

AMC10

10B

N/A

As shown in the figure below, six semicircles lie in the interior of a regular hexagon with side length 2 so that the diameters of the semicircles coincide with the sides of the hexagon. What is the area of the shaded region ---- inside the hexagon but outside all of the semicircles? [asy] size(140); fill((1,0)--(3,0)--(4,sqrt(3))--(3,2sqrt(3))--(1,2sqrt(3))--(0,sqrt(3))--cycle,gray(0.4)); fill(arc((2,0),1,180,0)--(2,0)--cycle,white); fill(arc((3.5,sqrt(3)/2),1,60,240)--(3.5,sqrt(3)/2)--cycle,white); fill(arc((3.5,3sqrt(3)/2),1,120,300)--(3.5,3sqrt(3)/2)--cycle,white); fill(arc((2,2sqrt(3)),1,180,360)--(2,2sqrt(3))--cycle,white); fill(arc((0.5,3sqrt(3)/2),1,240,420)--(0.5,3sqrt(3)/2)--cycle,white); fill(arc((0.5,sqrt(3)/2),1,300,480)--(0.5,sqrt(3)/2)--cycle,white); draw((1,0)--(3,0)--(4,sqrt(3))--(3,2sqrt(3))--(1,2sqrt(3))--(0,sqrt(3))--(1,0)); draw(arc((2,0),1,180,0)--(2,0)--cycle); draw(arc((3.5,sqrt(3)/2),1,60,240)--(3.5,sqrt(3)/2)--cycle); draw(arc((3.5,3sqrt(3)/2),1,120,300)--(3.5,3sqrt(3)/2)--cycle); draw(arc((2,2sqrt(3)),1,180,360)--(2,2sqrt(3))--cycle); draw(arc((0.5,3sqrt(3)/2),1,240,420)--(0.5,3sqrt(3)/2)--cycle); draw(arc((0.5,sqrt(3)/2),1,300,480)--(0.5,sqrt(3)/2)--cycle); label("$2$",(3.5,3sqrt(3)/2),NE); [/asy]

3\sqrt{3} - \pi

[asy] real x=sqrt(3); real y=2sqrt(3); real z=3.5; real a=x/2; real b=0.5; real c=3a; pair A, B, C, D, E, F; A = (1,0); B = (3,0); C = (4,x); D = (3,y); E = (1,y); F = (0,x); fill(A--B--C--D--E--F--A--cycle,grey); fill(arc((2,0),1,0,180)--cycle,white); fill(arc((2,y),1,180,360)--cycle,white); fill(arc((z,a),1,60,240)--cycle,white); fill(arc((b,a),1,300,480)--cycle,white); fill(arc((b,c),1,240,420)--cycle,white); fill(arc((z,c),1,120,300)--cycle,white); draw(A--B--C--D--E--F--A); draw(arc((z,c),1,120,300)); draw(arc((b,c),1,240,420)); draw(arc((b,a),1,300,480)); draw(arc((z,a),1,60,240)); draw(arc((2,y),1,180,360)); draw(arc((2,0),1,0,180)); label("2",(z,c),NE); pair X,Y,Z; X = (1,0); Y = (2,0); Z = (1.5,a); pair d = 1.9*dir(7); dot(X); dot(Y); dot(Z); label("A",X,SW); label("B",Y,SE); label("C",Z,N); draw(X--Y--Z--A); label("1",(1.5,0),S); label("1",(1.75,a/2),dir(30)); draw(anglemark(Y,X,Z,8),blue); label("$60^\circ$",anglemark(Y,X,Z),d); [/asy] Let point $A$ be a vertex of the regular hexagon, let point $B$ be the midpoint of the line connecting point $A$ and a neighboring vertex, and let point $C$ be the second intersection of the two semicircles that pass through point $A$. Then, $BC = 1$, since $B$ is the center of the semicircle with radius $1$ that $C$ lies on, $AB = 1$, since $B$ is the center of the semicircle with radius $1$ that $A$ lies on, and $\angle BAC = 60^\circ$, as a regular hexagon has angles of 120$^\circ$, and $\angle BAC$ is half of any angle in this hexagon. Now, using the Law of Sines, $\frac{1}{\sin \angle ACB} = \frac{1}{\sin 60^\circ}$, so $\angle ACB = 60^\circ$. Since the angles in a triangle sum to 180$^\circ$, $\angle ABC$ is also 60$^\circ$. Therefore, $\triangle ABC$ is an equilateral triangle with side lengths of $1$. [asy] real x=sqrt(3); real y=2sqrt(3); real z=3.5; real a=x/2; real b=0.5; real c=3a; pair A, B, C, D, E, F; A = (1,0); B = (3,0); C = (4,x); D = (3,y); E = (1,y); F = (0,x); fill(A--B--C--D--E--F--A--cycle,grey); fill(arc((2,0),1,0,180)--cycle,white); fill(arc((2,y),1,180,360)--cycle,white); fill(arc((z,a),1,60,240)--cycle,white); fill(arc((b,a),1,300,480)--cycle,white); fill(arc((b,c),1,240,420)--cycle,white); fill(arc((z,c),1,120,300)--cycle,white); draw(A--B--C--D--E--F--A); draw(arc((z,c),1,120,300)); draw(arc((b,c),1,240,420)); draw(arc((b,a),1,300,480)); draw(arc((z,a),1,60,240)); draw(arc((2,y),1,180,360)); draw(arc((2,0),1,0,180)); pair G,H,I,J,K; G = (2,0); H = (2.5,a); I = (1.5,a); J = (1,0); K = (3,0); pair d = 2.7*dir(78); dot(G); dot(H); dot(I); dot(J); dot(K); label("2",(z,c),NE); label("1",(1.5,0),S); label("1",(2.5,0),S); add(pathticks(G--J,1,0.5,0,3,red)); add(pathticks(I--J,1,0.5,0,3,red)); add(pathticks(G--I,1,0.5,0,3,red)); add(pathticks(G--H,1,0.5,0,3,red)); add(pathticks(G--K,1,0.5,0,3,red)); add(pathticks(K--H,1,0.5,0,3,red)); label("$60^\circ$",anglemark(H,G,I),d); draw(anglemark(H,G,I,8),blue); label("$60^\circ$",G,2*dir(146)); draw(anglemark(I,G,J,8),blue); label("$60^\circ$",G,2.8*dir(28)); draw(anglemark(K,G,H,8),blue); draw(G--J--I--G); draw(G--H--K--G); [/asy] Since the area of a regular hexagon can be found with the formula $\frac{3\sqrt{3}s^2}{2}$, where $s$ is the side length of the hexagon, the area of this hexagon is $\frac{3\sqrt{3}(2^2)}{2} = 6\sqrt{3}$. Since the area of an equilateral triangle can be found with the formula $\frac{\sqrt{3}}{4}s^2$, where $s$ is the side length of the equilateral triangle, the area of an equilateral triangle with side lengths of $1$ is $\frac{\sqrt{3}}{4}\left(1^2\right) = \frac{\sqrt{3}}{4}$. Since the area of a circle can be found with the formula $\pi r^2$, the area of a sixth of a circle with radius $1$ is $\frac{\pi(1^2)}{6} = \frac{\pi}{6}$. In each sixth of the hexagon, there are two equilateral triangles colored white, each with an area of $\frac{\sqrt{3}}{4}$, and one-sixth of a circle with radius $1$ colored white, with an area of $\frac{\pi}{6}$. The rest of the sixth is colored gray. Therefore, the total area that is colored white in each sixth of the hexagon is $2\left(\frac{\sqrt{3}}{4}\right) + \frac{\pi}{6}$, which equals $\frac{\sqrt{3}}{2} + \frac{\pi}{6}$, and the total area colored white is $6\left(\frac{\sqrt{3}}{2} + \frac{\pi}{6}\right)$, which equals $3\sqrt{3} + \pi$. Since the area colored gray equals the total area of the hexagon minus the area colored white, the area colored gray is $6\sqrt{3} - (3\sqrt{3} + \pi)$, which equals $\fbox{3\sqrt{3} - \pi}$.

AMC10 Second Half

AMC10 B

HMMT

HMMT-Nov

guts

Nov

Jeff has a 50 point quiz at $11 \mathrm{am}$. He wakes up at a random time between $10 \mathrm{am}$ and noon, then arrives at class 15 minutes later. If he arrives on time, he will get a perfect score, but if he arrives more than 30 minutes after the quiz starts, he will get a 0 , but otherwise, he loses a point for each minute he's late (he can lose parts of one point if he arrives a nonintegral number of minutes late). What is Jeff's expected score on the quiz?

\frac{55}{2}

If Jeff wakes up between 10:00 and 10:45, he gets 50 . If he wakes up between 10:45 and 11:15, and he wakes up $k$ minutes after 10:45, then he gets $50-k$ points. Finally, if he wakes up between 11:15 and 12:00 he gets 0 points. So he has a $\frac{3}{8}$ probability of 50 , a $\frac{3}{8}$ probability of 0 , and a $\frac{1}{4}$ probability of a number chosen uniformly between 20 and 50 (for an average of 35). Thus his expected score is $\frac{3}{8} \times 50+\frac{1}{4} \times 35=\frac{75+35}{4}=\frac{110}{4}=\frac{55}{2}$. $\fbox{\frac{55}{2}}$.

HMMT Nov Guts

HMMT-Nov Guts

HMMT

HMMT-Feb

team

Feb

The complex numbers $x, y, z$ satisfy \[ \begin{aligned} x y z & =-4 \\ (x+1)(y+1)(z+1) & =7 \\ (x+2)(y+2)(z+2) & =-3 \end{aligned} \] Find, with proof, the value of $(x+3)(y+3)(z+3)$.

-28

Solution 1. Consider the cubic polynomial $f(t)=(x+t)(y+t)(z+t)$. By the theory of finite differences, $f(3)-3 f(2)+3 f(1)-f(0)=3 !=6$, since $f$ is monic. Thus $f(3)=$ $6+3 f(2)-3 f(1)+f(0)=6+3(-3)-3(7)+(-4)=-28$. Solution 2. Alternatively, note that the system of equations is a (triangular) linear system in $w:=x y z$, $v:=x y+y z+z x$, and $u:=x+y+z$. The unique solution $(u, v, w)$ to this system is $\left(-\frac{27}{2}, \frac{47}{2},-4\right)$. Plugging in yields \[ \begin{aligned} (x+3)(y+3)(z+3) & =w+3 v+9 u+27 \\ & =-4+3 \cdot \frac{47}{2}+9 \cdot\left(-\frac{27}{2}\right)+27 \\ & =-28 \end{aligned} \] Remark. Since $f(0)<0, f(1)>0, f(2)<0, f(+\infty)>0$, the intermediate value theorem tells us the roots $-x,-y,-z$ of $f$ are real numbers in $(0,1),(1,2)$, and $(2,+\infty)$, in some order. With a little more calculation, one finds that $x, y, z$ are the three distinct zeroes of the polynomial $X^{3}+\frac{27}{2} X^{2}+\frac{47}{2} X+4$. The three zeroes are approximately $-11.484,-1.825$, and -0.191 . $\fbox{-28}$.

HMMT Feb Team

HMMT-Feb Team

AMC

AMC10

10B

N/A

Elmo makes $N$ sandwiches for a fundraiser. For each sandwich he uses $B$ globs of peanut butter at $4\cent$ per glob and $J$ blobs of jam at $5\cent$ per blob. The cost of the peanut butter and jam to make all the sandwiches is $$ 2.53$. Assume that $B$, $J$, and $N$ are positive integers with $N>1$. What is the cost of the jam Elmo uses to make the sandwiches?

1.65

The peanut butter and jam for each sandwich costs $4B\cent+5J\cent$, so the peanut butter and jam for $N$ sandwiches costs $N(4B+5J)\cent$. Setting this equal to $253\cent$: $N(4B+5J)=253=11\cdot23$ The only possible positive integer pairs $(N , 4B+5J)$ whose product is $253$ are: $(1,253) ; (11,23) ; (23,11) ; (253,1)$ The first pair violates $N>1$ and the third and fourth pairs have no positive integer solutions for $B$ and $J$. So, $N=11$ and $4B+5J=23$. The only integer solutions for $B$ and $J$ are $B=2$ and $J=3$. Therefore, the cost of the jam Elmo uses to make the sandwiches is $3\cdot5\cdot11=165\cent$ $= $1.65 \Rightarrow \fbox{1.65}$

AMC10 Final Problems

AMC10 B

AMC

AMC10

10A

N/A

Coin $A$ is flipped three times and coin $B$ is flipped four times. What is the probability that the number of heads obtained from flipping the two fair coins is the same?

\frac{35}{128}

There are $4$ ways that the same number of heads will be obtained; $0$, $1$, $2$, or $3$ heads. The probability of both getting $0$ heads is $\left(\frac12\right)^3{3\choose0}\left(\frac12\right)^4{4\choose0}=\frac1{128}$ The probability of both getting $1$ head is $\left(\frac12\right)^3{3\choose1}\left(\frac12\right)^4{4\choose1}=\frac{12}{128}$ The probability of both getting $2$ heads is $\left(\frac12\right)^3{3\choose2}\left(\frac12\right)^4{4\choose2}=\frac{18}{128}$ The probability of both getting $3$ heads is $\left(\frac12\right)^3{3\choose3}\left(\frac12\right)^4{4\choose3}=\frac{4}{128}$ Therefore, the probabiliy of flipping the same number of heads is: $\frac{1+12+18+4}{128}=\frac{35}{128}\Rightarrow\fbox{\frac{35}{128}}$

AMC10 First Half

AMC10 A

HMMT

HMMT-Nov

thm

Nov

For breakfast, Mihir always eats a bowl of Lucky Charms cereal, which consists of oat pieces and marshmallow pieces. He defines the luckiness of a bowl of cereal to be the ratio of the number of marshmallow pieces to the total number of pieces. One day, Mihir notices that his breakfast cereal has exactly 90 oat pieces and 9 marshmallow pieces, and exclaims, "This is such an unlucky bowl!" How many marshmallow pieces does Mihir need to add to his bowl to double its luckiness?

11

Let $x$ be the number of marshmallows to add. We are given that \[ 2 \cdot \frac{9}{99}=\frac{9+x}{99+x} \] Rearanging this gives \[ 2(99+x)=11(9+x) \] Thus $9 x=99$ and $x=11$. $\fbox{11}$.

HMMT Nov Easy

HMMT-Nov Theme

HMMT

HMMT-Feb

comb

Feb

A light pulse starts at a corner of a reflective square. It bounces around inside the square, reflecting off of the square's perimeter $n$ times before ending in a different corner. The path of the light pulse, when traced, divides the square into exactly 2021 regions. Compute the smallest possible value of $n$.

129

Solution: The main claim is that if the light pulse reflects vertically (on the left/right edges) $a$ times and horizontally $b$ times, then $\operatorname{gcd}(a+1, b+1)=1$, and the number of regions is $\frac{(a+2)(b+2)}{2}$. This claim can be conjectured by looking at small values of $a$ and $b$; we give a full proof at the end. Assuming the claim, we are trying to find the least possible value of $a+b$ when $(a+2)(b+2)=2 \cdot 2021=$ $2 \cdot 43 \cdot 47$. This happens when $(a+2, b+2)=(47,86)$, which also satisfies $\operatorname{gcd}(a+1, b+1)=1$, and gives $a+b=47+86-4=129$. We now prove the claim. Imagine that at each reflection, it is the square that gets reflected instead. Then the path $p$ of the light pulse becomes a straight segment $s$ from $(0,0)$ to $(a+1, b+1)$ of slope $+m=\frac{a+1}{b+1}$. \begin{itemize} \item The square starts as 1 region; the light pulse hitting a corner at the end creates 1 more region. \item Each reflection of the light pulse creates a region. These correspond to intersections of $s$ with a line $x=n$ or $y=n$ for $x \in[a], y \in[b]$. There are $a+b$ such intersections. \item Each self-intersection of $p$ creates a region. An intersection on $p$ corresponds to two on $s$, and each intersection of $s$ happens with a line of slope $-m$ passing through an even integral point, i.e. a line of the form $(b+1) x+(a+1) y=2 k$. The open segment $s$ intersects these lines for $k \in[a b+a+b]$. However, the $a+b$ intersections that happens on a gridline $x \in \mathbb{Z}$ or $y \in \mathbb{Z}$ do not count, so here we have an additional $a b / 2$ regions. \end{itemize} Therefore, the total number of regions is \[ 2+a+b+\frac{a b}{2}=\frac{(a+2)(b+2)}{2} \] $\fbox{129}$.

HMMT Feb Hard

HMMT-Feb Combinatorics

AMC

AMC10

10A

N/A

Two cubical dice each have removable numbers $1$ through $6$. The twelve numbers on the two dice are removed, put into a bag, then drawn one at a time and randomly reattached to the faces of the cubes, one number to each face. The dice are then rolled and the numbers on the two top faces are added. What is the probability that the sum is $7$?

\frac{2}{11}

At the moment when the numbers are in the bag, imagine that each of them has a different color. Clearly the situation is symmetric at this moment. Hence after we draw them, attach them and throw the dice, the probability of getting some pair of colors is the same for any two colors. There are ${12 \choose 2} = 66$ ways to pick two of the colors. We now have to count the ways where the two chosen numbers will have a sum of $7$. A sum of $7$ can be obtained as $1+6$, $2+5$, or $3+4$. Each number in the bag has two different colors, hence each of these three options corresponds to four pairs of colors. Out of the $66$ pairs of colors we can get when throwing the dice, $3\cdot 4=12$ will give us the sum $7$. Hence the probability that this will happen is $\frac{12}{66} = \fbox{\frac{2}{11}}$.

AMC10 Final Problems

AMC10 A

AIME

AIME

II

N/A

Define the sequence $a_1, a_2, a_3, \ldots$ by $a_n = \sum\limits_{k=1}^n \sin{k}$, where $k$ represents radian measure. Find the index of the 100th term for which $a_n < 0$.

628

If $n = 1$, $a_n = \sin(1) > 0$. Then if $n$ satisfies $a_n < 0$, $n \ge 2$, and \[a_n = \sum_{k=1}^n \sin(k) = \cfrac{1}{\sin{1}} \sum_{k=1}^n\sin(1)\sin(k) = \cfrac{1}{2\sin{1}} \sum_{k=1}^n\cos(k - 1) - \cos(k + 1) = \cfrac{1}{2\sin(1)} [\cos(0) + \cos(1) - \cos(n) - \cos(n + 1)].\] Since $2\sin 1$ is positive, it does not affect the sign of $a_n$. Let $b_n = \cos(0) + \cos(1) - \cos(n) - \cos(n + 1)$. Now since $\cos(0) + \cos(1) = 2\cos\left(\cfrac{1}{2}\right)\cos\left(\cfrac{1}{2}\right)$ and $\cos(n) + \cos(n + 1) = 2\cos\left(n + \cfrac{1}{2}\right)\cos\left(\cfrac{1}{2}\right)$, $b_n$ is negative if and only if $\cos\left(\cfrac{1}{2}\right) < \cos\left(n + \cfrac{1}{2}\right)$, or when $n \in [2k\pi - 1, 2k\pi]$. Since $\pi$ is irrational, there is always only one integer in the range, so there are values of $n$ such that $a_n < 0$ at $2\pi, 4\pi, \cdots$. Then the hundredth such value will be when $k = 100$ and $n = \lfloor 200\pi \rfloor = \lfloor 628.318 \rfloor = \fbox{628}$.

Very Hard AIME Problems

AIME

AMC

AMC12

12A

N/A

Square $ABCD$ has sides of length $4$, and $M$ is the midpoint of $\overline{CD}$. A circle with radius $2$ and center $M$ intersects a circle with radius $4$ and center $A$ at points $P$ and $D$. What is the distance from $P$ to $\overline{AD}$? [asy] pair A,B,C,D,M,P; D=(0,0); C=(10,0); B=(10,10); A=(0,10); M=(5,0); P=(8,4); dot(M); dot(P); draw(A--B--C--D--cycle,linewidth(0.7)); draw((5,5)..D--C..cycle,linewidth(0.7)); draw((7.07,2.93)..B--A--D..cycle,linewidth(0.7)); label("$A$",A,NW); label("$B$",B,NE); label("$C$",C,SE); label("$D$",D,SW); label("$M$",M,S); label("$P$",P,N); [/asy]

\frac {16}{5}

Let $D$ be the origin. $A$ is the point $(0,4)$ and $M$ is the point $(2,0)$. We are given the radius of the quarter circle and semicircle as $4$ and $2$, respectively, so their equations, respectively, are: $x^2 + (y-4)^2 = 4^2$ $(x-2)^2 + y^2 = 2^2$ Subtract the second equation from the first: $x^2 + (y - 4)^2 - (x - 2)^2 - y^2 = 12$ $4x - 8y + 12 = 12$ $x = 2y.$ Then substitute: $(2y)^2 + (y - 4)^2 = 16$ $4y^2 + y^2 - 8y + 16 = 16$ $5y^2 - 8y = 0$ $y(5y - 8) = 0.$ Thus $y = 0$ and $y = \frac{8}{5}$ making $x = 0$ and $x = \frac{16}{5}$. The first value of $0$ is obviously referring to the x-coordinate of the point where the circles intersect at the origin, $D$, so the second value must be referring to the x coordinate of $P$. Since $\overline{AD}$ is the y-axis, the distance to it from $P$ is the same as the x-value of the coordinate of $P$, so the distance from $P$ to $\overline{AD}$ is $\frac{16}{5} \Rightarrow \fbox{\frac {16}{5}}.$ -mathbeast9

AMC12 Second Half

AMC12 A

HMMT

HMMT-Feb

guts

Feb

Let $\omega$ be a circle, and let $A B C D$ be a quadrilateral inscribed in $\omega$. Suppose that $B D$ and $A C$ intersect at a point $E$. The tangent to $\omega$ at $B$ meets line $A C$ at a point $F$, so that $C$ lies between $E$ and $F$. Given that $A E=6, E C=4, B E=2$, and $B F=12$, find $D A$.

2\sqrt{42}

By power of a point, we have $E D \cdot E B=E A \cdot E C$, whence $E D=12$. Additionally, by power of a point, we have $144=F B^{2}=F C \cdot F A=F C(F C+10)$, so $F C=8$. Note that $\angle F B C=$ $\angle F A B$ and $\angle C F B=\angle A F B$, so $\triangle F B C \sim \triangle F A B$. Thus, $A B / B C=F A / F B=18 / 12=3 / 2$, so $A B=3 k$ and $B C=2 k$ for some $k$. Since $\triangle B E C \sim \triangle A E D$, we have $A D / B C=A E / B E=3$, so $A D=3 B C=6 k$. By Stewart's theorem on $\triangle E B F$, we have \[ (4)(8)(12)+(2 k)^{2}(12)=(2)^{2}(8)+(12)^{2}(4) \Longrightarrow 8+k^{2}=8 / 12+12 \] whence $k^{2}=14 / 3$. Thus, \[ D A=6 k=6 \sqrt{14 / 3}=6 \frac{\sqrt{42}}{3}=2 \sqrt{42} \] $\fbox{2\sqrt{42}}$.

HMMT Feb Guts

HMMT-Feb Guts

HMMT

HMMT-Feb

guts

Feb

You have 128 teams in a single elimination tournament. The Engineers and the Crimson are two of these teams. Each of the 128 teams in the tournament is equally strong, so during each match, each team has an equal probability of winning. Now, the 128 teams are randomly put into the bracket. What is the probability that the Engineers play the Crimson sometime during the tournament?

\frac{1}{64}

There are $\left(\begin{array}{c}128 \\ 2\end{array}\right)=127 \cdot 64$ pairs of teams. In each tournament, 127 of these pairs play. By symmetry, the answer is $\frac{127}{127 \cdot 64}=\frac{1}{64}$. $\fbox{\frac{1}{64}}$.

HMMT Feb Guts

HMMT-Feb Guts

HMMT

HMMT-Feb

guts

Feb

Let $f: \mathbb{Z} \rightarrow \mathbb{Z}$ be a function such that for any integers $x, y$, we have \[ f\left(x^{2}-3 y^{2}\right)+f\left(x^{2}+y^{2}\right)=2(x+y) f(x-y) \] Suppose that $f(n)>0$ for all $n>0$ and that $f(2015) \cdot f(2016)$ is a perfect square. Find the minimum possible value of $f(1)+f(2)$.

246

Plugging in $-y$ in place of $y$ in the equation and comparing the result with the original equation gives \[ (x-y) f(x+y)=(x+y) f(x-y) \] This shows that whenever $a, b \in \mathbb{Z}-\{0\}$ with $a \equiv b(\bmod 2)$, we have \[ \frac{f(a)}{a}=\frac{f(b)}{b} \] which implies that there are constants $\alpha=f(1) \in \mathbb{Z}_{>0}, \beta=f(2) \in \mathbb{Z}_{>0}$ for which $f$ satisfies the equation $(*)$ : \[ f(n)= \begin{cases}n \cdot \alpha & \text { when } 2 \nmid n \\ \frac{n}{2} \cdot \beta & \text { when } 2 \mid n\end{cases} \] Therefore, $f(2015) f(2016)=2015 \alpha \cdot 1008 \beta=2^{4} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 13 \cdot 31 \alpha \beta$, so $\alpha \beta=5 \cdot 7 \cdot 13 \cdot 31 \cdot t^{2}$ for some $t \in \mathbb{Z}_{>0}$. We claim that $(\alpha, \beta, t)=(5 \cdot 31,7 \cdot 13,1)$ is a triple which gives the minimum $\alpha+\beta$. In particular, we claim $\alpha+\beta \geq 246$. Consider the case $t \geq 2$ first. We have, by AM-GM, $\alpha+\beta \geq 2 \cdot \sqrt{\alpha \beta} \geq 4 \cdot \sqrt{14105}>246$. Suppose $t=1$. We have $\alpha \cdot \beta=5 \cdot 7 \cdot 13 \cdot 31$. Because $(\alpha+\beta)^{2}-(\alpha-\beta)^{2}=4 \alpha \beta$ is fixed, we want to have $\alpha$ as close as $\beta$ as possible. This happens when one of $\alpha, \beta$ is $5 \cdot 31$ and the other is $7 \cdot 13$. In this case, $\alpha+\beta=91+155=246$. Finally, we note that the equality $f(1)+f(2)=246$ can be attained. Consider $f: \mathbb{Z} \rightarrow \mathbb{Z}$ such that $f(n)=91 n$ for every odd $n \in \mathbb{Z}$ and $f(n)=\frac{155}{2} n$ for every even $n \in \mathbb{Z}$. It can be verified that $f$ satisfies the condition in the problem and $f(1)+f(2)=246$ as claimed. $\fbox{246}$.

HMMT Feb Guts

HMMT-Feb Guts

AMC

AMC10

10B

N/A

The first four terms in an arithmetic sequence are $x+y$, $x-y$, $xy$, and $\frac{x}{y}$, in that order. What is the fifth term?

\frac{123}{40}

The difference between consecutive terms is $(x-y)-(x+y)=-2y.$ Therefore we can also express the third and fourth terms as $x-3y$ and $x-5y.$ Then we can set them equal to $xy$ and $\frac{x}{y}$ because they are the same thing. \begin{align} xy&=x-3y\\ xy-x&=-3y\\ x(y-1)&=-3y\\ x&=\frac{-3y}{y-1} \end{align} Substitute into our other equation. \[\frac{x}{y}=x-5y\] \[\frac{-3}{y-1}=\frac{-3y}{y-1}-5y\] \[-3=-3y-5y(y-1)\] \[0=5y^2-2y-3\] \[0=(5y+3)(y-1)\] \[y=-\frac35, 1\] But $y$ cannot be $1$ because then the first term would be $x+1$ and the second term $x-1$ while the last two terms would be equal to $x.$ Therefore $y=-\frac35.$ Substituting the value for $y$ into any of the equations, we get $x=-\frac98.$ Finally, \[\frac{x}{y}-2y=\frac{9\cdot 5}{8\cdot 3}+\frac{6}{5}=\fbox{\frac{123}{40}}\]

AMC10 Final Problems

AMC10 B

HMMT

HMMT-Feb

guts

Feb

Let $\varphi(n)$ denote the number of positive integers less than or equal to $n$ which are relatively prime to $n$. Let $S$ be the set of positive integers $n$ such that $\frac{2 n}{\varphi(n)}$ is an integer. Compute the sum \[ \sum_{n \in S} \frac{1}{n} \]

\frac{10}{3}

Solution: Let $T_{n}$ be the set of prime factors of $n$. Then \[ \frac{2 n}{\phi(n)}=2 \prod_{p \in T} \frac{p}{p-1} \] We can check that this is an integer for the following possible sets: \[ \varnothing,\{2\},\{3\},\{2,3\},\{2,5\},\{2,3,7\} \] For each set $T$, the sum of the reciprocals of the positive integers having that set of prime factors is \[ \prod_{p \in T}\left(\sum_{m=1}^{\infty} \frac{1}{p^{m}}\right)=\prod_{p \in T} \frac{1}{p-1} \] Therefore the desired sum is \[ 1+1+\frac{1}{2}+\frac{1}{2}+\frac{1}{4}+\frac{1}{12}=\frac{10}{3} \] $\fbox{\frac{10}{3}}$.

HMMT Feb Guts

HMMT-Feb Guts

AMC

AMC8

8

N/A

Isabella must take four 100-point tests in her math class. Her goal is to achieve an average grade of 95 on the tests. Her first two test scores were 97 and 91. After seeing her score on the third test, she realized she can still reach her goal. What is the lowest possible score she could have made on the third test?

92

Isabella wants an average grade of $95$ on her 4 tests; this also means that she wants the sum of her test scores to be at least $95 \times 4 = 380$ (if she goes over this number, she'll be over her goal!). She's already taken two tests, which sum to $97+91 = 188$, which means she needs $192$ more points to achieve her desired average. In order to minimize the score on the third test, we assume that Isabella will receive all $100$ points on the fourth test. Therefore, the lowest Isabella could have scored on the third test would be $192-100 = \fbox{92}$.

AMC8 First Half

AMC8

AIME

AIME

I

N/A

Freddy the frog is jumping around the coordinate plane searching for a river, which lies on the horizontal line $y = 24$. A fence is located at the horizontal line $y = 0$. On each jump Freddy randomly chooses a direction parallel to one of the coordinate axes and moves one unit in that direction. When he is at a point where $y=0$, with equal likelihoods he chooses one of three directions where he either jumps parallel to the fence or jumps away from the fence, but he never chooses the direction that would have him cross over the fence to where $y < 0$. Freddy starts his search at the point $(0, 21)$ and will stop once he reaches a point on the river. Find the expected number of jumps it will take Freddy to reach the river.

273

Clearly Freddy's $x$-coordinate is irrelevant, so we let $E(y)$ be the expected value of the number of jumps it will take him to reach the river from a given $y$-coordinate. Observe that $E(24)=0$, and \[E(y)=1+\frac{E(y+1)+E(y-1)+2E(y)}{4}\] for all $y$ such that $1\le y\le 23$. Also note that $E(0)=1+\frac{2E(0)+E(1)}{3}$. This gives $E(0)=E(1)+3$. Plugging this into the equation for $E(1)$ gives that \[E(1)=1+\frac{E(2)+3E(1)+3}{4},\] or $E(1)=E(2)+7$. Iteratively plugging this in gives that $E(n)=E(n+1)+4n+3$. Thus $E(23)=E(24)+95$, $E(22)=E(23)+91=E(24)+186$, and $E(21)=E(22)+87=E(24)+273=\fbox{273}$.

Very Hard AIME Problems

AIME

HMMT

HMMT-Feb

comb

Feb

In the figure below, how many ways are there to select 5 bricks, one in each row, such that any two bricks in adjacent rows are adjacent?

61

The number of valid selections is equal to the number of paths which start at a top brick and end at a bottom brick. We compute these by writing 1 in each of the top bricks and letting lower bricks be the sum of the one or two bricks above them. Thus, the number inside each brick is the number of paths from that brick to the top. The bottom row is $6,14,16,15,10$, which sums to 61 . \begin{center} \begin{tabular}{|c|c|c|c|c|c|c|} \hline 1 & 1 & 1 & 1 & 1 & 1 & \\ \hline & 2 & 2 & $\frac{1}{2}$ & 2 & & 1 \\ \hline 2 & 4 & 4 & 4 & 4 & 3 & \\ \hline & 6 & 8 & 8 & 7 & & 3 \\ \hline 6 & 14 & 16 & 15 & & 10 & \\ \hline \end{tabular} \end{center} $\fbox{61}$.

HMMT Feb Easy

HMMT-Feb Combinatorics

HMMT

HMMT-Nov

team

Nov

There is a unit circle that starts out painted white. Every second, you choose uniformly at random an arc of arclength 1 of the circle and paint it a new color. You use a new color each time, and new paint covers up old paint. Let $c_{n}$ be the expected number of colors visible after $n$ seconds. Compute $\lim _{n \rightarrow \infty} c_{n}$.

4 \pi

Solution 1: Notice that colors always appear in contiguous arcs on the circle (i.e. there's never a color that appears in two disconnected arcs). So the number of distinct visible colors is equal to the number of radii that serve as boundaries between colors. Each time we place a new color, we create 2 more of these radii, but all of the previous radii have a $p=\frac{1}{2 \pi}$ chance of being covered up. It is well-known that, given an event that occurs with probability $p$, it takes an expected $\frac{1}{p}$ trials for it to happen - this means that any given radii has an expected lifespan of $2 \pi$ before it gets covered up (i.e. it remains visible for $2 \pi$ turns on average). Thus, over the course of $n \gg 1$ turns there are $2 n$ total radii created, each lasting an average of $2 \pi$ turns, so there are $\frac{4 \pi n}{n}=4 \pi$ radii visible on average each turn. A more rigorous way to see this is the two radii created on the most recent turn have a probability 1 of being exposed; the two radii created last turn each have a probability $1-p$ of being exposed; the two radii created two turns ago each have a probability $(1-p)^{2}$ of being exposed, and so on. Thus, on turn $n$, the expected number of exposed radii is \[ 2\left(1+(1-p)+(1-p)^{2}+\cdots+(1-p)^{n-1}\right) \] This geometric series converges to $\frac{2}{p}$ as $n$ grows. Solution 2: Notice that colors always appear in contiguous arcs on the circle (i.e. there's never a color that appears in two disconnected arcs). So the number of distinct visible colors is equal to the number of radii that serve as boundaries between colors. Each time we place a new color, we create 2 more of these radii. However, if the expected number of colors after turn $n$ is constant, we must expect to remove 2 of these radii each turn. This is only possible when there are $4 \pi$ total radii, since we expect to remove $\frac{1}{2 \pi}$ of them each time. Solution 3: Consider the probability that the $k$-th last added arc is visible. Suppose there are $j$ arcs after the $k$-th last arc that partially covers this arc. Then the probability that the $k$-th last arc is still visible is $\frac{j+1}{2^{j}}$, since this is equivalent to randomly choosing $j$ positions within the $k$-th last arc to place an arc in, then randomly choosing a direction, and there are $2^{j}$ ways to choose directions and $j+1$ of them are good. The probability that any arc partially covers the $k$-th last arc is $\frac{2}{2 \pi}$. Putting everything together, the probability that the $k$-th last arc is visible is \[ \sum_{j=0}^{k-1} \frac{j+1}{2^{j}} \cdot\left(\begin{array}{c} k-1 \\ j \end{array}\right) \cdot\left(\frac{2}{2 \pi}\right)^{j} \cdot\left(1-\frac{2}{2 \pi}\right)^{k-1-j} \] so the answer is \[ 1+\sum_{k=2}^{n} \sum_{j=0}^{k-1} \frac{j+1}{2^{j}} \cdot\left(\begin{array}{c} k-1 \\ j \end{array}\right) \cdot\left(\frac{2}{2 \pi}\right)^{j} \cdot\left(1-\frac{2}{2 \pi}\right)^{k-1-j} \] (as the last arc is definitely visible). We can write this as \[ 1+\sum_{k=2}^{n} \sum_{j=0}^{k-1}(j+1) \cdot\left(\begin{array}{c} k-1 \\ j \end{array}\right) \cdot\left(\frac{1}{2 \pi}\right)^{j} \cdot\left(1-\frac{2}{2 \pi}\right)^{k-1-j} \] Now, \[ \sum_{j=0}^{k-1} 1 \cdot\left(\begin{array}{c} k-1 \\ j \end{array}\right) \cdot\left(\frac{1}{2 \pi}\right)^{j} \cdot\left(1-\frac{2}{2 \pi}\right)^{k-1-j}=\left(\frac{1}{2 \pi}+1-\frac{2}{2 \pi}\right)^{k-1} \] by binomial theorem. We can write $j \cdot\left(\begin{array}{c}k-1 \\ j\end{array}\right)=(k-1)\left(\begin{array}{c}k-2 \\ j-1\end{array}\right)$, so \[ \begin{aligned} \sum_{j=0}^{k-1} j \cdot\left(\begin{array}{c} k-1 \\ j \end{array}\right) \cdot\left(\frac{1}{2 \pi}\right)^{j} \cdot\left(1-\frac{2}{2 \pi}\right)^{k-1-j} & =(k-1) \sum_{j=1}^{k-1}\left(\begin{array}{c} k-2 \\ j-1 \end{array}\right) \cdot\left(\frac{1}{2 \pi}\right)^{j} \cdot\left(1-\frac{2}{2 \pi}\right)^{k-1-j} \\ & =(k-1) \sum_{j=0}^{k-2}\left(\begin{array}{c} k-2 \\ j \end{array}\right) \cdot\left(\frac{1}{2 \pi}\right)^{j+1} \cdot\left(1-\frac{2}{2 \pi}\right)^{k-2-j} \\ & =\frac{k-1}{2 \pi} \cdot\left(\frac{1}{2 \pi}+1-\frac{2}{2 \pi}\right)^{k-2} \end{aligned} \] Therefore the answer is \[ 1+\sum_{k=2}^{n}\left(1-\frac{1}{2 \pi}+\frac{k-1}{2 \pi}\right)\left(1-\frac{1}{2 \pi}\right)^{k-2} \] which is an arithmetic sequence times a geometric sequence. Standard techniques simplify this to $4 \pi$. Solution 4: We solve for the lifespan of an arc. Let $f(x)$ represent the expected number of turns an arc of length $2 \pi x$ will remain visible. Our final answer will be to calculate $f\left(\frac{1}{2 \pi}\right)$ Then we get the recurrence \[ f(x)=1+\left(\frac{2 \pi-1}{2 \pi}-x\right) f(x)+2 \int_{0}^{x} f(x) d x \] The 1 term comes from counting the fact that the arc is visible during the current turn. The $\left(\frac{2 \pi-1}{2 \pi}-\right.$ $x) f(x)$ term comes from the fact that there is a $\frac{2 \pi-1}{2 \pi}-x$ chance that the next arc will not intersect the current arc, in which case the current arc would get an extra $f(x)$ turns to live. The integral comes from the fact that we want to take the average of $f(y)$ for $y \sim_{r}[0, x]$, which corresponds to the next arc covering up $2 \pi(x-y)$ of the current one. If we differentiate both sides, we end up with a differential equation. Solve it via separation. We end up with $f(x)=4 \pi^{2} x+2 \pi$. Plugging in $x=\frac{1}{2 \pi}$ gets us $4 \pi$. In general, the answer is $f(x)=\frac{x}{k^{2}}+\frac{1}{k}$, where we are placing arcs of length $2 \pi k$. $\fbox{4 \pi}$.

HMMT Nov Team

HMMT-Nov Team

AMC

AMC10

10A

N/A

Simplify $\sqrt[3]{x\sqrt[3]{x\sqrt[3]{x\sqrt{x}}}}$.

\sqrt{x}

$\sqrt[3]{x\sqrt{x}}=\sqrt[3]{(\sqrt{x})^{2}\cdot\sqrt{x}}=\sqrt[3]{(\sqrt{x})^{3}}=\sqrt{x}$. Therefore: $\sqrt[3]{x\sqrt[3]{x\sqrt[3]{x\sqrt{x}}}}=\sqrt[3]{x\sqrt[3]{x\sqrt{x}}}=\sqrt[3]{x\sqrt{x}}= \sqrt{x} \Rightarrow \fbox{\sqrt{x}}$

AMC10 First Half

AMC10 A

HMMT

HMMT-Nov

team

Nov

Mario is once again on a quest to save Princess Peach. Mario enters Peach's castle and finds himself in a room with 4 doors. This room is the first in a sequence of 2 indistinugishable rooms. In each room, 1 door leads to the next room in the sequence (or, for the second room, into Bowser's level), while the other 3 doors lead to the first room. Suppose that in every room, Mario randomly picks a door to walk through. What is the expected number of doors (not including Mario's initial entrance to the first room) through which Mario will pass before he reaches Bowser's level?

20

Let $E_{i}$ be the expected number of doors through which Mario will pass in the future if he is currently in room $i$ for $i=1,2,3$ (we will set $E_{3}=0$ ). We claim that $E_{i}=1+\frac{3}{4} E_{1}+\frac{1}{4} E_{i+1}$. Indeed, the 1 at the beginning comes from the fact that we need to pass through a door to leave the room, the ${ }_{4}^{3} E_{1}$ comes from the fact that there is a $\frac{3}{4}$ chance of ending up in room 1 , and the $\frac{1}{4} E_{i+1}$ corresponds to the fact that there is a $\frac{1}{4}$ chance of ending up in $E_{i+1}$. Using this, we get $E_{1}=1+\frac{3}{4} E_{1}+\frac{1}{4} E_{2}$, or $E_{1}=4+E_{2}$. We also get $E_{2}=1+\frac{3}{4} E_{1}$. Solving this system of equations yields $E_{1}=20$. $\fbox{20}$.

HMMT Nov Team

HMMT-Nov Team

AIME

AIME

II

N/A

A rectangle has sides of length $a$ and 36. A hinge is installed at each vertex of the rectangle, and at the midpoint of each side of length 36. The sides of length $a$ can be pressed toward each other keeping those two sides parallel so the rectangle becomes a convex hexagon as shown. When the figure is a hexagon with the sides of length $a$ parallel and separated by a distance of 24, the hexagon has the same area as the original rectangle. Find $a^2$. [asy] pair A,B,C,D,E,F,R,S,T,X,Y,Z; dotfactor = 2; unitsize(.1cm); A = (0,0); B = (0,18); C = (0,36); // don't look here D = (12*2.236, 36); E = (12*2.236, 18); F = (12*2.236, 0); draw(A--B--C--D--E--F--cycle); dot(" ",A,NW); dot(" ",B,NW); dot(" ",C,NW); dot(" ",D,NW); dot(" ",E,NW); dot(" ",F,NW); //don't look here R = (12*2.236 +22,0); S = (12*2.236 + 22 - 13.4164,12); T = (12*2.236 + 22,24); X = (12*4.472+ 22,24); Y = (12*4.472+ 22 + 13.4164,12); Z = (12*4.472+ 22,0); draw(R--S--T--X--Y--Z--cycle); dot(" ",R,NW); dot(" ",S,NW); dot(" ",T,NW); dot(" ",X,NW); dot(" ",Y,NW); dot(" ",Z,NW); // sqrt180 = 13.4164 // sqrt5 = 2.236[/asy]

720

When we squish the rectangle, the hexagon is composed of a rectangle and two isosceles triangles with side lengths 18, 18, and 24 as shown below. [asy] pair R,S,T,X,Y,Z; dotfactor = 2; unitsize(.1cm); R = (12*2.236 +22,0); S = (12*2.236 + 22 - 13.4164,12); T = (12*2.236 + 22,24); X = (12*4.472+ 22,24); Y = (12*4.472+ 22 + 13.4164,12); Z = (12*4.472+ 22,0); draw(R--S--T--X--Y--Z--cycle); draw(T--R,red); draw(X--Z,red); dot(" ",R,NW); dot(" ",S,NW); dot(" ",T,NW); dot(" ",X,NW); dot(" ",Y,NW); dot(" ",Z,NW); // sqrt180 = 13.4164 // sqrt5 = 2.236[/asy] By Heron's Formula, the area of each isosceles triangle is $\sqrt{(30)(12)(12)(6)}=\sqrt{180\times 12^2}=72\sqrt{5}$. So the area of both is $144\sqrt{5}$. From the rectangle, our original area is $36a$. The area of the rectangle in the hexagon is $24a$. So we have \[24a+144\sqrt{5}=36a\implies 12a=144\sqrt{5}\implies a=12\sqrt{5}\implies a^2=\fbox{720}.\]

Easy AIME Problems

AIME

HMMT

HMMT-Feb

guts

Feb

Rosencrantz and Guildenstern play a game in which they repeatedly flip a fair coin. Let $a_{1}=4$, $a_{2}=3$, and $a_{n}=a_{n-1}+a_{n-2}$ for all $n \geq 3$. On the $n$th flip, if the coin is heads, Rosencrantz pays Guildenstern $a_{n}$ dollars, and, if the coin is tails, Guildenstern pays Rosencrantz $a_{n}$ dollars. If play continues for 2010 turns, what is the probability that Rosencrantz ends up with more money than he started with?

\frac{1}{2}-\frac{1}{2^{1341}}

Since Rosencrantz and Guildenstern have an equal chance of winning each toss, both have the same probability of ending up with a positive amount of money. Let $x$ denote the probability that they both end up with zero dollars. We wish to find $\frac{1-x}{2}$. We have $x$ is equal to the probability that \[ s_{2010}:=i_{1} a_{1}+i_{2} a_{2}+\cdots i_{2010} a_{2010}=0 \] where $i_{n}$ has an equal probability of being either 1 or -1 . We claim that $s_{2010}=0$ if and only if $i_{3 n}=-i_{3 n-1}=-i_{3 n-2}$ for all $n \leq 670$. We start with the following lemma. Lemma. We have $a_{n}>\sum_{k=1}^{n-3} a_{k}$ for all $n \geq 4$. Proof: For the case $n=4, a_{4}=a_{3}+a_{2}=2 a_{2}+a_{1}>a_{1}$. In case $n>4$, we have \[ a_{n}=a_{n-2}+a_{n-1}>a_{n-2}+\sum_{k=1}^{n-4} a_{k}=a_{n-4}+\sum_{k=1}^{n-3} a_{k}>\sum_{k=1}^{n-3} a_{k} \] It suffices to show that $s_{3 n}=0$ only if $i_{3 k}=-i_{3 k-1}=-i_{3 k-2}$ for all $k \leq n$. The triangle inequality implies the following: \[ \begin{aligned} & 0 \leq|| i_{3 n-2} a_{3 n-2}+s_{3 n-3}|-| i_{3 n-1} a_{3 n-1}+i_{3 n} a_{3 n}|| \leq\left|s_{3 n}\right|=0 \\ & 0 \leq|| i_{3 n-1} a_{3 n-1}+s_{3 n-3}|-| i_{3 n-2} a_{3 n-2}+i_{3 n} a_{3 n}|| \leq\left|s_{3 n}\right|=0 \end{aligned} \] By the lemma, we have \[ \begin{array}{rll} a_{3 n-2}+\sum_{k=1}^{3 n-3} a_{k}<a_{3 n-2}+a_{3 n} & < & a_{3 n-1}+a_{3 n} \\ a_{3 n-1}+\sum_{k=1}^{3 n-3} a_{k}<a_{3 n-1}+a_{3 n-3}+a_{3 n-4}+a_{3 n-2} & = & a_{3 n-2}+a_{3 n} \end{array} \] $i_{3 n}=i_{3 n-1}$ implies $\left|a_{3 n-2}+\sum_{k=1}^{3 n-3} a_{k}\right|<\left|i_{3 n-1} a_{3 n-1}+i_{3 n} a_{3 n}\right|$ and $i_{3 n}=i_{3 n-2}$ implies $\mid a_{3 n-1}+$ $\sum_{k=1}^{3 n-3} a_{k}|<| i_{3 n-2} a_{3 n-2}+i_{3 n} a_{3 n} \mid$, which are both contradictions; therefore, we must have $i_{3 n}=-i_{3 n-1}$ and $i_{3 n}=-i_{3 n-2}$. The probability that $i_{3 n}=-i_{3 n-1}=-i_{3 n-2}$ is $\frac{1}{4}$, so $x=\left(\frac{1}{4}\right)^{670}=\frac{1}{2^{1340}}$, and $\frac{1-x}{2}=\frac{1}{2}-\frac{1}{2^{1341}}$. $\fbox{\frac{1}{2}-\frac{1}{2^{1341}}}$.

HMMT Feb Guts

HMMT-Feb Guts

AMC

AMC8

8

N/A

Ms. Hamilton’s eighth-grade class wants to participate in the annual three-person-team basketball tournament. Lance, Sally, Joy, and Fred are chosen for the team. In how many ways can the three starters be chosen?

4

There are $\binom{4}{3}$ ways to choose three starters. Thus the answer is $\fbox{4}$.

AMC8 First Half

AMC8

AMC

AMC10

10B

N/A

Define the operation $\star$ by $a \star b = (a+b)b.$ What is $(3 \star 5) - (5 \star 3)?$

16

Substitute and simplify. \[(3+5)5 - (5+3)3 = (3+5)2 = 8\cdot2 = \fbox{16}\]

AMC10 First Half

AMC10 B

HMMT

HMMT-Nov

team

Nov

Marcus and four of his relatives are at a party. Each pair of the five people are either friends or enemies. For any two enemies, there is no person that they are both friends with. In how many ways is this possible?

52

Denote friendship between two people $a$ and $b$ by $a \sim b$. Then, assuming everyone is friends with themselves, the following conditions are satisfied: \begin{itemize} \item $a \sim a$ \item If $a \sim b$, then $b \sim a$ \item If $a \sim b$ and $b \sim c$, then $a \sim c$ \end{itemize} Thus we can separate the five people into a few groups (possibly one group), such that people are friends within each group, but two people are enemies when they are in different groups. Here comes the calculation. Since the number of group(s) can be 1,2,3,4, or 5, we calculate for each of those cases. When there's only one group, then we only have 1 possibility that we have a group of 5 , and the total number of friendship assignments in this case is $\left(\begin{array}{l}5 \\ 5\end{array}\right)=1$; when there are two groups, we have $5=1+4=2+3$ are all possible numbers of the two groups, with a total of $\left(\begin{array}{l}5 \\ 1\end{array}\right)+\left(\begin{array}{l}5 \\ 2\end{array}\right)=15$ choices; when there are three groups, then we have $5=1+1+3=1+2+2$, with $\left(\begin{array}{l}5 \\ 3\end{array}\right)+\frac{\left(\begin{array}{l}5 \\ 1\end{array}\right)\left(\begin{array}{l}5 \\ 2\end{array}\right)}{2}=25$ possibilities; when there are four of them, then we have $5=1+1+1+2$ be its only possibility, with $\left(\begin{array}{l}5 \\ 2\end{array}\right)=10$ separations; when there are 5 groups, obviously we have 1 possibility. Hence, we have a total of $1+15+25+10+1=52$ possibilities. Alternatively, we can also solve the problem recursively. Let $B_{n}$ be the number of friendship graphs with $n$ people, and consider an arbitrary group. If this group has size $k$, then there are $\left(\begin{array}{l}n \\ k\end{array}\right)$ possible such groups, and $B_{n-k}$ friendship graphs on the remaining $n-k$ people. Therefore, we have the recursion \[ B_{n}=\sum_{k=0}^{n}\left(\begin{array}{l} n \\ k \end{array}\right) B_{n-k} \] with the initial condition $B_{1}=1$. Calculating routinely gives $B_{5}=52$ as before. $\fbox{52}$.

HMMT Nov Team

HMMT-Nov Team

HMMT

HMMT-Feb

comb

Feb

The country of HMMTLand has 8 cities. Its government decides to construct several two-way roads between pairs of distinct cities. After they finish construction, it turns out that each city can reach exactly 3 other cities via a single road, and from any pair of distinct cities, either exactly 0 or 2 other cities can be reached from both cities by a single road. Compute the number of ways HMMTLand could have constructed the roads.

875

Solution: Let the cities be numbered $1,2,3,4,5,6,7,8$. WLOG, 1 is connected to 2,3 , and 4 . First suppose 2 and 3 are connected; then 3 and 1 share a second common neighbor, which must be 4 (as 1 is not connected to anything else). Likewise 2 and 4 are connected, and so 5, 6, 7, 8 are pairwise connected as well, so the graph consists of two disjoint copies of $K_{4}$ :\\ There are $\frac{1}{2}\left(\begin{array}{l}8 \\ 4\end{array}\right)=35$ ways to partition the 8 vertices into two groups of 4 , so there are 35 such graphs. Otherwise, none of 2, 3, 4 are connected to each other. Then 2 and 3 must share a common neighbor, as must 3 and 4 , and 2 and 4 . If these are the same neighbor, this vertex would share all three neighbors with 1 , so they must be pairwise distinct. The last vertex must then be connected to these three, creating a cube graph. A cube has 48 symmetries, so the number of such graphs is $\frac{8 !}{48}=840$. The total is $35+840=875$. $\fbox{875}$.

HMMT Feb Easy

HMMT-Feb Combinatorics

AMC

AMC8

8

N/A

A ball is dropped from a height of $3$ meters. On its first bounce it rises to a height of $2$ meters. It keeps falling and bouncing to $\frac{2}{3}$ of the height it reached in the previous bounce. On which bounce will it not rise to a height of $0.5$ meters?

5

Each bounce is $2/3$ times the height of the previous bounce. The first bounce reaches $2$ meters, the second $4/3$, the third $8/9$, the fourth $16/27$, and the fifth $32/81$. Half of $81$ is $40.5$, so the ball does not reach the required height on bounce $\fbox{5}$.

AMC8 First Half

AMC8

HMMT

HMMT-Feb

geo

Feb

In triangle $A B C, A B=2, A C=1+\sqrt{5}$, and $\angle C A B=54^{\circ}$. Suppose $D$ lies on the extension of $A C$ through $C$ such that $C D=\sqrt{5}-1$. If $M$ is the midpoint of $B D$, determine the measure of $\angle A C M$, in degrees.

63

Let $E$ be the midpoint of $\overline{A D} . E C=\sqrt{5}+1-\sqrt{5}=1$, and $E M=1$ by similar triangles $(A B D \sim E M D) . \triangle E C M$ is isosceles, with $m \angle C E M=54^{\circ}$. Thus $m \angle A C M=m \angle E C M=$ $\frac{180-54}{2}=63^{\circ}$. $\fbox{63}$.

HMMT Feb Hard

HMMT-Feb Geometry

HMMT

HMMT-Feb

guts

Feb

Let $n$ be a positive integer, and let $s$ be the sum of the digits of the base-four representation of $2^{n}-1$. If $s=2023$ (in base ten), compute $n$ (in base ten).

1349

Solution: Every power of 2 is either represented in base 4 as $100 \ldots 00_{4}$ or $200 . .00_{4}$ with some number of zeros. That means every positive integer in the form $2^{n}-1$ is either represented in base 4 as $333 \ldots 33_{4}$ or $133 \ldots 33$ for some number threes. Note that $2023=2022+1=674 \cdot 3+1$, meaning $2^{n}-1$ must be $133 \ldots 333_{4}$ with 674 threes. Converting this to base 2 results in \[ 133 \ldots 33_{4}=200 \ldots 00_{4}-1=2 \cdot 4^{674}-1=2^{1349}-1 \] for an answer of 1349. $\fbox{1349}$.

HMMT Feb Guts

HMMT-Feb Guts

AMC

AMC12

12B

N/A

In $\triangle ABC$, $\angle C = 90^\circ$ and $AB = 12$. Squares $ABXY$ and $CBWZ$ are constructed outside of the triangle. The points $X$, $Y$, $Z$, and $W$ lie on a circle. What is the perimeter of the triangle?

12+12\sqrt{2}

[asy] pair A,B,C,M,E,W,Z,X,Y; A=(2,0); B=(0,2); C=(0,0); M=(A+B)/2; W=(-2,2); Z=(-2,-0); X=(2,4); Y=(4,2); E=(W+Z)/2; draw(A--B--C--cycle); draw(W--B--C--Z--cycle); draw(A--B--X--Y--cycle); dot(M); dot(E); label("W",W,NW); label("Z",Z,SW); label("C",C,S); label("A",A,S); label("B",B,N); label("X",X,NE); label("Y",Y,SE); label("E",E,1.5*plain.W); label("M",M,NE); draw(circle(M,sqrt(10))); [/asy] First, we should find the center and radius of this circle. We can find the center by drawing the perpendicular bisectors of $WZ$ and $XY$ and finding their intersection point. This point happens to be the midpoint of $AB$, the hypotenuse. Let this point be $M$. To find the radius, determine $MY$, where $MY^{2} = MA^2 + AY^2$, $MA = \frac{12}{2} = 6$, and $AY = AB = 12$. Thus, the radius $=r =MY = 6\sqrt5$. Next we let $AC = b$ and $BC = a$. Consider the right triangle $ACB$ first. Using the Pythagorean theorem, we find that $a^2 + b^2 = 12^2 = 144$. [asy] pair A,B,C,M,E,W,Z,X,Y; A=(2,0); B=(0,2); C=(0,0); M=(A+B)/2; W=(-2,2); Z=(-2,-0); X=(2,4); Y=(4,2); E=(W+Z)/2; draw(A--B--C--cycle); draw(W--B--C--Z--cycle); draw(A--B--X--Y--cycle); dot(M); dot(E); label("W",W,NW); label("Z",Z,SW); label("C",C,S); label("A",A,S); label("B",B,N); label("X",X,NE); label("Y",Y,SE); label("E",E,1.5*plain.W); label("M",M,NE); draw(circle(M,sqrt(10))); draw(E--Z--M--cycle,dashed); [/asy] Now, we let $E$ be the midpoint of $WZ$, and we consider right triangle $ZEM$. By the Pythagorean theorem, we have that $\left(\frac{a}{2}\right)^2 + \left(a + \frac{b}{2}\right)^2 = r^2 = 180$. Expanding this equation, we get that \[\frac{1}{4}(a^2+b^2) + a^2 + ab = 180\] \[\frac{144}{4} + a^2 + ab = 180\] \[a^2 + ab = 144 = a^2 + b^2\] \[ab = b^2\] \[b = a\] This means that $ABC$ is a 45-45-90 triangle, so $a = b = \frac{12}{\sqrt2} = 6\sqrt2$. Thus the perimeter is $a + b + AB = 12\sqrt2 + 12$ which is answer $\fbox{12+12\sqrt{2}}$.

AMC12 Second Half

AMC12 B

HMMT

HMMT-Nov

guts

Nov

Define the annoyingness of a permutation of the first $n$ integers to be the minimum number of copies of the permutation that are needed to be placed next to each other so that the subsequence $1,2, \ldots, n$ appears. For instance, the annoyingness of $3,2,1$ is 3 , and the annoyingness of $1,3,4,2$ is 2 . A random permutation of $1,2, \ldots, 2022$ is selected. Compute the expected value of the annoyingness of this permutation.

\frac{2023}{2}

Solution: For a given permutation $p_{1}, \ldots, p_{n}$, let $f_{k}(p)$ be the smallest number of copies of $p$ that need to be placed next to each other to have $1, \ldots, k$ appear as a subsequence. We are interested in finding the expectation of $f_{n}(p)$. Notice that if $k$ appears before $k+1$ in $p$, then $f_{k}(p)=f_{k+1}(p)$. Otherwise, $f_{k}(p)+1=f_{k+1}(p)$. Since $f_{1}(p)$ is always 1 , this tells us that $f_{n}(p)$ is equal to 1 plus the number of values of $k$ that exist such that $k+1$ appears before $k$. But for any such $k$, this occurs with probability $1 / 2$. By linearity of expectation, the answer is $1+2021 / 2=\frac{2023}{2}$. $\fbox{\frac{2023}{2}}$.

HMMT Nov Guts

HMMT-Nov Guts

AMC

AMC12

12A

N/A

The vertices of a $3-4-5$ right triangle are the centers of three mutually externally tangent circles, as shown. What is the sum of the areas of the three circles?

14\pi

Let the radius of the smallest circle be $r_A$, the radius of the second largest circle be $r_B$, and the radius of the largest circle be $r_C$. \[r_A + r_B = 3\] \[r_A + r_C = 4\] \[r_ B + r_C = 5\] Adding up all these equations and then dividing both sides by 2, we get, \[r_A + r_B + r_C = 6\] Then, we get $r_A = 1$, $r_B = 2$, and $r_C = 3$ Then we get $1^2 \pi + 2^2 \pi + 3^2 \pi = 14 \pi \iff$ $\fbox{14\pi}$.

AMC12 Second Half

AMC12 A

HMMT

HMMT-Nov

gen

Nov

Let $P$ and $Q$ be points on line $l$ with $P Q=12$. Two circles, $\omega$ and $\Omega$, are both tangent to $l$ at $P$ and are externally tangent to each other. A line through $Q$ intersects $\omega$ at $A$ and $B$, with $A$ closer to $Q$ than $B$, such that $A B=10$. Similarly, another line through $Q$ intersects $\Omega$ at $C$ and $D$, with $C$ closer to $Q$ than $D$, such that $C D=7$. Find the ratio $A D / B C$.

\frac{8}{9}

We first apply the Power of a Point theorem repeatedly. Note that $Q A \cdot Q B=Q P^{2}=$ $Q C \cdot Q D$. Substituting in our known values, we obtain $Q A(Q A+10)=12^{2}=Q C(Q C+7)$. Solving these quadratics, we get that $Q A=8$ and $Q C=9$. We can see that $\frac{A Q}{D Q}=\frac{C Q}{B Q}$ and that $\angle A Q D=\angle C Q B$, so $Q A D \sim Q C B$. (Alternatively, going back to the equality $Q A \cdot Q B=Q C \cdot Q D$, we realize that this is just a Power of a Point theorem on the quadrilateral $A B D C$, and so this quadrilateral is cyclic. This implies that $\angle A D Q=\angle A D C=$ $\angle A B C=\angle Q B C$.) Thus, $\frac{A D}{B C}=\frac{A Q}{Q C}=\frac{8}{9}$. $\fbox{\frac{8}{9}}$.

HMMT Nov Hard

HMMT-Nov General

HMMT

HMMT-Feb

geo

Feb

Let $A B C$ be a triangle with circumradius $R=17$ and inradius $r=7$. Find the maximum possible value of $\sin \frac{A}{2}$.

\frac{17+\sqrt{51}}{34}

Letting $I$ and $O$ denote the incenter and circumcenter of triangle $A B C$ we have by the triangle inequality that \[ A O \leq A I+O I \Longrightarrow R \leq \frac{r}{\sin \frac{A}{2}}+\sqrt{R(R-2 r)} \] and by plugging in our values for $r$ and $R$ we get \[ \sin \frac{A}{2} \leq \frac{17+\sqrt{51}}{34} \] as desired. Equality holds when $A B C$ is isosceles and $I$ lies between $A$ and $O$. $\fbox{\frac{17+\sqrt{51}}{34}}$.

HMMT Feb Hard

HMMT-Feb Geometry

HMMT

HMMT-Nov

gen

Nov

Rosencrantz plays $n \leq 2015$ games of question, and ends up with a win rate (i.e. \# of games won ) of $k$. Guildenstern has also played several games, and has a win rate less than $k$. He realizes that if, after playing some more games, his win rate becomes higher than $k$, then there must have been some point in time when Rosencrantz and Guildenstern had the exact same win-rate. Find the product of all possible values of $k$.

\frac{1}{2015}

Write $k=\frac{m}{n}$, for relatively prime integers $m, n$. For the property not to hold, there must exist integers $a$ and $b$ for which \[ \frac{a}{b}<\frac{m}{n}<\frac{a+1}{b+1} \] (i.e. at some point, Guildenstern must "jump over" $k$ with a single win) \[ \Longleftrightarrow a n+n-m>b m>a n \] hence there must exist a multiple of $m$ strictly between $a n$ and $a n+n-m$. If $n-m=1$, then the property holds as there is no integer between $a n$ and $a n+n-m=a n+1$. We now show that if $n-m \neq 1$, then the property does not hold. By Bzout's Theorem, as $n$ and $m$ are relatively prime, there exist $a$ and $x$ such that $a n=m x-1$, where $0<a<m$. Then $a n+n-m \geq a n+2=m x+1$, so $b=x$ satisfies the conditions. As a result, the only possible $k$ are those in the form $\frac{n}{n+1}$. We know that Rosencrantz played at most 2015 games, so the largest non-perfect winrate he could possibly have is $\frac{2014}{2015}$. Therefore, $k \in\left\{\frac{1}{2}, \frac{2}{3}, \ldots, \frac{2014}{2015}\right\}$, the product of which is $\frac{1}{2015}$. $\fbox{\frac{1}{2015}}$.

HMMT Nov Hard

HMMT-Nov General

AMC

AMC10

10B

N/A

What is the greatest power of $2$ that is a factor of $10^{1002} - 4^{501}$?

2^{1005}

First, we can write the expression in a more primitive form which will allow us to start factoring. \[10^{1002} - 4^{501} = 2^{1002} \cdot 5^{1002} - 2^{1002}\] Now, we can factor out $2^{1002}$. This leaves us with $5^{1002} - 1$. Call this number $N$. Thus, our final answer will be $2^{1002+k}$, where $k$ is the largest power of $2$ that divides $N$. Now we can consider $N \pmod{16}$, since $k \le 4$ by the answer choices. Note that \begin{align} 5^1 &\equiv 5 \pmod{16} \\ 5^2 &\equiv 9 \pmod{16} \\ 5^3 &\equiv 13 \pmod{16} \\ 5^4 &\equiv 1 \pmod{16} \\ 5^5 &\equiv 5 \pmod{16} \\ &\: \: \qquad \vdots \end{align} The powers of $5$ cycle in $\mod{16}$ with a period of $4$. Thus, \[5^{1002} \equiv 5^2 \equiv 9 \pmod{16} \implies 5^{1002} - 1 \equiv 8 \pmod{16}\] This means that $N$ is divisible by $8= 2^3$ but not $16 = 2^4$, so $k = 3$ and our answer is $2^{1002 + 3} =\fbox{2^{1005}}$.

AMC10 Second Half

AMC10 B

HMMT

HMMT-Feb

guts

Feb

Find the number of pairs of integers $(x, y)$ such that $x^{2}+2 y^{2}<25$.

55

We do casework on $y$. If $y=0$, we have $x^{2}<25$, so we get 9 values of $x$. If $y= \pm 1$, then $x^{2}<23$, so we still have 9 values of $x$. If $y= \pm 2$, we have $x^{2}<17$, so we have 9 values of $x$. If $y= \pm 3$, we have $x^{2}<7$, we get 5 values of $x$. Therefore, the final answer is $9+2(9+9+5)=55$. $\fbox{55}$.

HMMT Feb Guts

HMMT-Feb Guts

AIME

AIME

II

N/A

The teams $T_1$, $T_2$, $T_3$, and $T_4$ are in the playoffs. In the semifinal matches, $T_1$ plays $T_4$, and $T_2$ plays $T_3$. The winners of those two matches will play each other in the final match to determine the champion. When $T_i$ plays $T_j$, the probability that $T_i$ wins is $\frac{i}{i+j}$, and the outcomes of all the matches are independent. The probability that $T_4$ will be the champion is $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.

781

There are two scenarios in which $T_4$ wins. The first scenario is where $T_4$ beats $T_1$, $T_3$ beats $T_2$, and $T_4$ beats $T_3$, and the second scenario is where $T_4$ beats $T_1$, $T_2$ beats $T_3$, and $T_4$ beats $T_2$. Consider the first scenario. The probability $T_4$ beats $T_1$ is $\frac{4}{4+1}$, the probability $T_3$ beats $T_2$ is $\frac{3}{3+2}$, and the probability $T_4$ beats $T_3$ is $\frac{4}{4+3}$. Therefore the first scenario happens with probability $\frac{4}{4+1}\cdot\frac{3}{3+2}\cdot\frac{4}{4+3}$. Consider the second scenario. The probability $T_4$ beats $T_1$ is $\frac{4}{1+4}$, the probability $T_2$ beats $T_3$ is $\frac{2}{2+3}$, and the probability $T_4$ beats $T_2$ is $\frac{4}{4+2}$. Therefore the second scenario happens with probability $\frac{4}{1+4}\cdot\frac{2}{2+3}\cdot\frac{4}{4+2}$. By summing these two probabilities, the probability that $T_4$ wins is $\frac{4}{4+1}\cdot\frac{3}{3+2}\cdot\frac{4}{4+3}+\frac{4}{1+4}\cdot\frac{2}{2+3}\cdot\frac{4}{4+2}$. Because this expression is equal to $\frac{256}{525}$, the answer is $256+525=\fbox{781}$.

Easy AIME Problems

AIME

AMC

AMC8

8

N/A

Gilda has a bag of marbles. She gives $20\%$ of them to her friend Pedro. Then Gilda gives $10\%$ of what is left to another friend, Ebony. Finally, Gilda gives $25\%$ of what is now left in the bag to her brother Jimmy. What percentage of her original bag of marbles does Gilda have left for herself?

54

After Gilda gives $20$% of the marbles to Pedro, she has $80$% of the marbles left. If she then gives $10$% of what's left to Ebony, she has $(0.8*0.9)$ = $72$% of what she had at the beginning. Finally, she gives $25$% of what's left to her brother, so she has $(0.75*0.72)$ $\fbox{54}$ of what she had in the beginning left.

AMC8 First Half

AMC8

HMMT

HMMT-Feb

alg

Feb

Let $\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right),\left(x_{3}, y_{3}\right),\left(x_{4}, y_{4}\right)$, and $\left(x_{5}, y_{5}\right)$ be the vertices of a regular pentagon centered at $(0,0)$. Compute the product of all positive integers $k$ such that the equality \[ x_{1}^{k}+x_{2}^{k}+x_{3}^{k}+x_{4}^{k}+x_{5}^{k}=y_{1}^{k}+y_{2}^{k}+y_{3}^{k}+y_{4}^{k}+y_{5}^{k} \] must hold for all possible choices of the pentagon.

1152

Solution: Without loss of generality let the vertices of the pentagon lie on the unit circle. Then, if $f(\theta)=\cos (\theta)^{k}$ and $g(\theta)=\sum_{j=0}^{4} f(\theta+2 j \pi / 5)$, the condition becomes $g(\theta)=g(\pi / 2-\theta)$, or $g(\theta)=$ $g(\theta+\pi / 2)$, since $g$ is an odd function. Write $f \asymp g$ if $f=c g$ for some nonzero constant $c$ that we don't care about. Since $\cos \theta \asymp e^{i \theta}+e^{-i \theta}$, we find that \[ f(\theta) \asymp \sum_{\ell \in \mathbb{Z}}\left(\begin{array}{c} k \\ \frac{k+\ell}{2} \end{array}\right) e^{i \ell \theta} \] where $\left(\begin{array}{l}a \\ b\end{array}\right)$ is defined to be zero if $b$ is not an integer in the interval $[0, a]$. It is also true that \[ \sum_{j=0}^{4} e^{i \ell(\theta+2 j \pi / 5)}= \begin{cases}5 e^{i \theta} & 5 \mid \ell \\ 0 & \text { else }\end{cases} \] so \[ g(\theta) \asymp \sum_{\ell \in 5 \mathbb{Z}}\left(\begin{array}{c} k \\ \frac{k+\ell}{2} \end{array}\right) e^{i \ell \theta} \] This is periodic with period $\pi / 2$ if and only if all terms with $\ell$ not a multiple of 4 are equal to 0 . However, we know that the nonzero terms are exactly the $\ell$ that (1) are multiples of $5,(2)$ are of the same parity as $k$, and (3) satisfy $|\ell| \leq k$. Hence, if $k$ is even, the condition is satisfied if and only if $k<10$ (else the $\ell=10$ term is nonzero), and if $k$ is odd, the condition is satisfied if and only if $k<5$ (else the $\ell=5$ term is nonzero). Our final answer is $1 \cdot 2 \cdot 3 \cdot 4 \cdot 6 \cdot 8=1152$. $\fbox{1152}$.

HMMT Feb Hard

HMMT-Feb Algebra

AIME

AIME

I

N/A

Circles $\omega_1$ and $\omega_2$ intersect at points $X$ and $Y$. Line $\ell$ is tangent to $\omega_1$ and $\omega_2$ at $A$ and $B$, respectively, with line $AB$ closer to point $X$ than to $Y$. Circle $\omega$ passes through $A$ and $B$ intersecting $\omega_1$ again at $D \neq A$ and intersecting $\omega_2$ again at $C \neq B$. The three points $C$, $Y$, $D$ are collinear, $XC = 67$, $XY = 47$, and $XD = 37$. Find $AB^2$.

270

Let $Z = XY \cap AB$. By the radical axis theorem $AD, XY, BC$ are concurrent, say at $P$. Moreover, $\triangle DXP \sim \triangle PXC$ by simple angle chasing. Let $y = PX, x = XZ$. Then \[\frac{y}{37} = \frac{67}{y} \qquad \implies \qquad y^2 = 37 \cdot 67.\] Now, $AZ^2 = \tfrac 14 AB^2$, and by power of a point, \begin{align} x(y-x) &= \tfrac 14 AB^2, \quad \text{and} \\ x(47+x) &= \tfrac 14 AB^2 \end{align} Solving, we get \[\tfrac 14 AB^2 = \tfrac 12 (y-47)\cdot \tfrac 12 (y+47) \qquad \implies\] \[\qquad AB ^ 2 = 37\cdot 67 - 47^2 = \fbox{270}\]

Very Hard AIME Problems

AIME

HMMT

HMMT-Nov

gen

Nov

Emily's broken clock runs backwards at five times the speed of a regular clock. Right now, it is displaying the wrong time. How many times will it display the correct time in the next 24 hours? It is an analog clock (i.e. a clock with hands), so it only displays the numerical time, not AM or PM. Emily's clock also does not tick, but rather updates continuously.

12

Solution: When comparing Emily's clock with a normal clock, the difference between the two times decreases by 6 seconds for every 1 second that passes. Since this difference is treated as 0 whenever it is a multiple of 12 hours, the two clocks must agree once every $\frac{12}{6}=2$ hours. Thus, in a 24 hour period it will agree 12 times. $\fbox{12}$.

HMMT Nov Easy

HMMT-Nov General

HMMT

HMMT-Nov

thm

Nov

Omkar, $\mathrm{Krit}_{1}, \mathrm{Krit}_{2}$, and $\mathrm{Krit}_{3}$ are sharing $x>0$ pints of soup for dinner. Omkar always takes 1 pint of soup (unless the amount left is less than one pint, in which case he simply takes all the remaining soup). Krit $_{1}$ always takes $\frac{1}{6}$ of what is left, Krit $_{2}$ always takes $\frac{1}{5}$ of what is left, and $\mathrm{Krit}_{3}$ always takes $\frac{1}{4}$ of what is left. They take soup in the order of Omkar, $\mathrm{Krit}_{1}, \mathrm{Krit}_{2}, \mathrm{Krit}_{3}$, and then cycle through this order until no soup remains. Find all $x$ for which everyone gets the same amount of soup.

\frac{49}{3}

The main observation is that if $x>1$ pints of soup are left, then in one round, Omkar gets 1 and each Krit $_{n}$ gets $\frac{x-1}{6}$, with $\frac{x-1}{2}$ soup left. Thus it is evident that each Krit $_{n}$ gets the same amount of soup, which means it suffices to find $x$ for which Omkar gets $\frac{x}{4}$. Omkar gets 1 for each cycle and then all the remaining soup when there is less than one pint remaining. The amount of soup becomes (after each cycle) \[ x \rightarrow \frac{x-1}{2} \rightarrow \frac{x-3}{4} \rightarrow \cdots \rightarrow \frac{x+1}{2^{n}}-1 \] so if $n$ is the number of cycles, then Omkar's soup is $n+\frac{x+1}{2^{n}}-1$. Setting this equal to $\frac{x}{4}$, we obtain \[ x=\frac{n+1 / 2^{n}-1}{1 / 4-1 / 2^{n}}=\frac{(n-1) 2^{n}+1}{2^{n-2}-1} \] This immediately implies $n>2$. On the other hand, we necessarily have $0 \leq \frac{x+1}{2^{n}}-1 \leq 1$, so $2^{n} \leq x+1 \leq 2^{n+1}$. But \[ x+1=\frac{(n-1) 2^{n}+2^{n-2}}{2^{n-2}-1} \leq \frac{(n-1) 2^{n}+2^{n}}{2^{n-3}}=8 n \] So $2^{n} \leq 8 n \Longrightarrow n \leq 5$. Testing $n=3,4,5$ : \begin{itemize} \item For $n=3$ we get $x=17$ which is greater than $2^{4}$. \item For $n=4$ we get $x=\frac{49}{3}$ which works. \item For $n=5$ we get $x=\frac{129}{7}$ which is less than $2^{5}$. \end{itemize} We see that only $n=4$ and $x=\frac{49}{3}$ works. $\fbox{\frac{49}{3}}$.

HMMT Nov Hard

HMMT-Nov Theme

AMC

AMC10

10B

N/A

What is the product of all the roots of the equation \[\sqrt{5 | x | + 8} = \sqrt{x^2 - 16}.\]

-64

First, square both sides, and isolate the absolute value. \begin{align} 5|x|+8&=x^2-16\\ 5|x|&=x^2-24\\ |x|&=\frac{x^2-24}{5}. \\ \end{align} Solve for the absolute value and factor. Case 1: $x=\frac{x^2-24}{5}$ Multiplying both sides by $5$ gives us \[5x=x^2-24.\] Rearranging and factoring, we have \begin{align} x^2-5x-24 &=0, \\ (x-8)(x+3) &= 0.\\ \end{align} Case 2: $x=\frac{-x^2+24}{5}$ As above, we multiply both sides by $5$ to find \[5x=-x^2+24.\] Rearranging and factoring gives us \begin{align} x^2+5x-24 &=0, \\ (x+8)(x-3) &= 0. \\ \end{align} Combining these cases, we have $x= -8, -3, 3, 8$. Because our first step of squaring is not reversible, however, we need to check for extraneous solutions. Plug each solution for $x$ back into the original equation to ensure it works. Whether the number is positive or negative does not matter since the absolute value or square will cancel it out anyways. Trying $|x|=|3|$, we have \begin{align} \sqrt{5|3|+8}&=\sqrt{3^2-16}, \\ \sqrt{15+8}&=\sqrt{9-16}, \\ \sqrt{23} &\not= \sqrt{-7}.\\ \end{align} Therefore, $x = 3$ and $x= -3$ are extraneous. Checking $|x|=|8|$, we have \begin{align} \sqrt{5|8|+8}&=\sqrt{8^2-16}, \\ \sqrt{40+8}&=\sqrt{64-16}, \\ \sqrt{48}&=\sqrt{48}.\\ \end{align} The roots of our original equation are $-8$ and $8$ and product is $-8 \times 8 = \fbox{-64}$.

AMC10 Second Half

AMC10 B

AMC

AMC12

12A

N/A

The area of a pizza with radius $4$ is $N$ percent larger than the area of a pizza with radius $3$ inches. What is the integer closest to $N$?

78

The area of the larger pizza is $16\pi$, while the area of the smaller pizza is $9\pi$. Therefore, the larger pizza is $\frac{7\pi}{9\pi} \cdot 100\%$ bigger than the smaller pizza. $\frac{7\pi}{9\pi} \cdot 100\% = 77.777....$, which is closest to $\fbox{78}$.

AMC12 First Half

AMC12 A

HMMT

HMMT-Nov

gen

Nov

Equilateral triangle $A B C$ has circumcircle $\Omega$. Points $D$ and $E$ are chosen on minor arcs $A B$ and $A C$ of $\Omega$ respectively such that $B C=D E$. Given that triangle $A B E$ has area 3 and triangle $A C D$ has area 4, find the area of triangle $A B C$.

\frac{37}{7}

A rotation by $120^{\circ}$ about the center of the circle will take $A B E$ to $B C D$, so $B C D$ has area 3 . Let $A D=x, B D=y$, and observe that $\angle A D C=\angle C D B=60^{\circ}$. By Ptolemy's Theorem, $C D=x+y$. We have \[ \begin{aligned} & 4=[A C D]=\frac{1}{2} A D \cdot C D \cdot \sin 60^{\circ}=\frac{\sqrt{3}}{4} x(x+y) \\ & 3=[B C D]=\frac{1}{2} B D \cdot C D \cdot \sin 60^{\circ}=\frac{\sqrt{3}}{4} y(x+y) \end{aligned} \] By dividing these equations find $x: y=4: 3$. Let $x=4 t, y=3$. Substitute this into the first equation to get $1=\frac{\sqrt{3}}{4} \cdot 7 t^{2}$. By the Law of Cosines, \[ A B^{2}=x^{2}+x y+y^{2}=37 t^{2} \] The area of $A B C$ is then \[ \frac{A B^{2} \sqrt{3}}{4}=\frac{37}{7} \] $\fbox{\frac{37}{7}}$.

HMMT Nov Hard

HMMT-Nov General

AMC

AMC12

12A

N/A

A power boat and a raft both left dock $A$ on a river and headed downstream. The raft drifted at the speed of the river current. The power boat maintained a constant speed with respect to the river. The power boat reached dock $B$ downriver, then immediately turned and traveled back upriver. It eventually met the raft on the river 9 hours after leaving dock $A.$ How many hours did it take the power boat to go from $A$ to $B$?

4.5

WLOG let the speed of the river be 0. This is allowed because the problem never states that the speed of the current has to have a magnitude greater than 0. In this case, when the powerboat travels from $A$ to $B$, the raft remains at $A$. Thus the trip from $A$ to $B$ takes the same time as the trip from $B$ to the raft. Since these times are equal and sum to $9$ hours, the trip from $A$ to $B$ must take half this time, or $4.5$ hours. The answer is thus $\fbox{4.5}$. Remark: This is equivalent to viewing the problem from the raft's perspective.

AMC12 Second Half

AMC12 A

AMC

AMC8

8

N/A

George walks $1$ mile to school. He leaves home at the same time each day, walks at a steady speed of $3$ miles per hour, and arrives just as school begins. Today he was distracted by the pleasant weather and walked the first $\frac{1}{2}$ mile at a speed of only $2$ miles per hour. At how many miles per hour must George run the last $\frac{1}{2}$ mile in order to arrive just as school begins today?

6

Using the harmonic mean formula, and making the speed he needs to take to get to school for the last half $x$, we can make the expression: $\frac{2 * 2x}{2 + x}$ Which simplifies to: $\frac{4x}{2 + x}$ Then, because we know that since he goes at $3$ mph on a normal day, we can say that it is the harmonic mean of the two rates he goes at today, so we add that to our expression and turn it into something familiar: $\frac{4x}{2 + x} = 3$ Solving that equation gives us: $\fbox{6}$

AMC8 Second Half

AMC8

HMMT

HMMT-Feb

geo

Feb

Circles $\omega_{1}$ and $\omega_{2}$ intersect at points $A$ and $B$. Segment $P Q$ is tangent to $\omega_{1}$ at $P$ and to $\omega_{2}$ at $Q$, and $A$ is closer to $P Q$ than $B$. Point $X$ is on $\omega_{1}$ such that $P X \| Q B$, and point $Y$ is on $\omega_{2}$ such that $Q Y \| P B$. Given that $\angle A P Q=30^{\circ}$ and $\angle P Q A=15^{\circ}$, find the ratio $A X / A Y$.

2-\sqrt{3}

Let $C$ be the fourth vertex of parallelogram $A P C Q$. The midpoint $M$ of $\overline{P Q}$ is the intersection of the diagonals of this parallelogram. Because $M$ has equal power ${ }^{3}$ with respect to the two circles $\omega_{1}$ and $\omega_{2}$, it lies on $\overleftrightarrow{A B}$, the circles' radical axi: $\stackrel{4}{4}$. Therefore, $C$ lies on $\overleftrightarrow{A B}$ as well. Using a series of parallel lines and inscribed arcs, we have: \[ \angle A P C=\angle A P Q+\angle C P Q=\angle A P Q+\angle P Q A=\angle A B P+\angle Q B A=\angle P B Q=\angle X P B \] where the last equality follows from the fact that $P X \| Q B$. We also know that $\angle B X P=180^{\circ}-\angle P A B=\angle C A P$, so triangles $B X P$ and $C A P$ are similar. By the spiral similarity theorem ${ }^{5}$ triangles $B P C$ and $X P A$ are similar, too. By analogous reasoning, triangles $B Q C$ and $Y Q A$ are similar. Then we have: \[ \frac{A X}{A Y}=\frac{A X / B C}{A Y / B C}=\frac{A P / C P}{A Q / C Q}=\frac{A P^{2}}{A Q^{2}} \] where the last inequality holds because $A P C Q$ is a parallelogram. Using the Law of Sines, the last expression equals $\frac{\sin ^{2} 15^{\circ}}{\sin ^{2} 30^{\circ}}=2-\sqrt{3}$. $\fbox{2-\sqrt{3}}$.

HMMT Feb Hard

HMMT-Feb Geometry

HMMT

HMMT-Feb

comb

Feb

Let $S$ be a set of intervals defined recursively as follows: \begin{itemize} Initially, $[1,1000]$ is the only interval in $S$. If $l \neq r$ and $[l, r] \in S$, then both $\left[l,\left\lfloor\frac{l+r}{2}\right\rfloor\right],\left[\left\lfloor\frac{l+r}{2}\right\rfloor+1, r\right] \in S$. \end{itemize} (Note that $S$ can contain intervals such as [1,1], which contain a single integer.) An integer $i$ is chosen uniformly at random from the range $[1,1000]$. What is the expected number of intervals in $S$ which contain $i$ ?

10.976

Solution: The answer is given by computing the sum of the lengths of all intervals in $S$ and dividing this value by 1000 , where the length of an interval $[i, j]$ is given by $j-i+1$. An interval may be categorized based on how many times $[1,1000]$ must be split to attain it. An interval that is derived from splitting $[1,1000] k$ times will be called a $k$-split. The only 0 -split is $[1,1000]$, with a total length of 1000 . The 1 -splits are $[1,500]$ and $[501,1000]$, with a total length of 1000. As long as none of the $k$-splits have length 1, the $(k+1)$-splits will have the same total length. Since the length of the intervals is reduced by half each time (rounded down), we find that the sum of the lengths of the $k$-splits is 1000 for $0 \leq k \leq 9$. Note that the 9-splits consist of $2^{10}-1000$ intervals of length 1 and $1000-2^{9}$ intervals of length 2. Then the 10 -splits consist of $2\left(1000-2^{9}\right)$ intervals of length 1 , with total length $2\left(1000-2^{9}\right)$. The total interval length across all splits is equal to $12(1000)-2^{10}$, so our answer is \[ 12-\frac{2^{10}}{1000}=10.976 \] $\fbox{10.976}$.

HMMT Feb Easy

HMMT-Feb Combinatorics

AMC

AMC10

10B

N/A

Let $\lfloor x \rfloor$ denote the greatest integer less than or equal to $x$. How many real numbers $x$ satisfy the equation $x^2 + 10,000\lfloor x \rfloor = 10,000x$?

199

This rewrites itself to $x^2=10,000\{x\}$ where $\lfloor x \rfloor + \{x\} = x$. Graphing $y=10,000\{x\}$ and $y=x^2$ we see that the former is a set of line segments with slope $10,000$ from $0$ to $1$ with a hole at $x=1$, then $1$ to $2$ with a hole at $x=2$ etc. Here is a graph of $y=x^2$ and $y=16\{x\}$ for visualization. [asy] import graph; size(400); xaxis("$x$",Ticks(Label(fontsize(8pt)),new real[]{-5,-4,-3, -2, -1,0,1 2,3, 4,5})); yaxis("$y$",Ticks(Label(fontsize(8pt)),new real[]{0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18})); real y(real x) {return x^2;} draw(circle((-4,16), 0.1)); draw(circle((-3,16), 0.1)); draw(circle((-2,16), 0.1)); draw(circle((-1,16), 0.1)); draw(circle((0,16), 0.1)); draw(circle((1,16), 0.1)); draw(circle((2,16), 0.1)); draw(circle((3,16), 0.1)); draw(circle((4,16), 0.1)); draw((-5,0)--(-4,16), black); draw((-4,0)--(-3,16), black); draw((-3,0)--(-2,16), black); draw((-2,0)--(-1,16), black); draw((-1,0)--(-0,16), black); draw((0,0)--(1,16), black); draw((1,0)--(2,16), black); draw((2,0)--(3,16), black); draw((3,0)--(4,16), black); draw(graph(y,-4.2,4.2),green); [/asy] Now notice that when $x=\pm 100$ the graph has a hole at $(\pm 100,10,000)$ which the equation $y=x^2$ passes through and then continues upwards. Thus our set of possible solutions is bounded by $(-100,100)$. We can see that $y=x^2$ intersects each of the lines once and there are $99-(-99)+1=199$ lines for an answer of $\fbox{199}$.

AMC10 Final Problems

AMC10 B

AIME

AIME

I

N/A

David found four sticks of different lengths that can be used to form three non-congruent convex cyclic quadrilaterals, $A,\text{ }B,\text{ }C$, which can each be inscribed in a circle with radius $1$. Let $\varphi_A$ denote the measure of the acute angle made by the diagonals of quadrilateral $A$, and define $\varphi_B$ and $\varphi_C$ similarly. Suppose that $\sin\varphi_A=\tfrac{2}{3}$, $\sin\varphi_B=\tfrac{3}{5}$, and $\sin\varphi_C=\tfrac{6}{7}$. All three quadrilaterals have the same area $K$, which can be written in the form $\dfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

59

Suppose our four sides lengths cut out arc lengths of $2a$, $2b$, $2c$, and $2d$, where $a+b+c+d=180^\circ$. Then, we only have to consider which arc is opposite $2a$. These are our three cases, so \[\varphi_A=a+c\] \[\varphi_B=a+b\] \[\varphi_C=a+d\] Our first case involves quadrilateral $ABCD$ with $\overarc{AB}=2a$, $\overarc{BC}=2b$, $\overarc{CD}=2c$, and $\overarc{DA}=2d$. Then, by Law of Sines, $AC=2\sin\left(\frac{\overarc{ABC}}{2}\right)=2\sin(a+b)$ and $BD=2\sin\left(\frac{\overarc{BCD}}{2}\right)=2\sin(a+d)$. Therefore, \[K=\frac{1}{2}\cdot AC\cdot BD\cdot \sin(\varphi_A)=2\sin\varphi_A\sin\varphi_B\sin\varphi_C=\frac{24}{35},\] so our answer is $24+35=\fbox{59}$. Note that the conditions of the problem are satisfied when the lengths of the four sticks are about $0.32, 0.91, 1.06, 1.82$. By S.B.

Very Hard AIME Problems

AIME

AMC

AMC10

10A

N/A

A rug is made with three different colors as shown. The areas of the three differently colored regions form an arithmetic progression. The inner rectangle is one foot wide, and each of the two shaded regions is $1$ foot wide on all four sides. What is the length in feet of the inner rectangle? [asy] size(6cm); defaultpen(fontsize(9pt)); path rectangle(pair X, pair Y){ return X--(X.x,Y.y)--Y--(Y.x,X.y)--cycle; } filldraw(rectangle((0,0),(7,5)),gray(0.5)); filldraw(rectangle((1,1),(6,4)),gray(0.75)); filldraw(rectangle((2,2),(5,3)),white); label("$1$",(0.5,2.5)); draw((0.3,2.5)--(0,2.5),EndArrow(TeXHead)); draw((0.7,2.5)--(1,2.5),EndArrow(TeXHead)); label("$1$",(1.5,2.5)); draw((1.3,2.5)--(1,2.5),EndArrow(TeXHead)); draw((1.7,2.5)--(2,2.5),EndArrow(TeXHead)); label("$1$",(4.5,2.5)); draw((4.5,2.7)--(4.5,3),EndArrow(TeXHead)); draw((4.5,2.3)--(4.5,2),EndArrow(TeXHead)); label("$1$",(4.1,1.5)); draw((4.1,1.7)--(4.1,2),EndArrow(TeXHead)); draw((4.1,1.3)--(4.1,1),EndArrow(TeXHead)); label("$1$",(3.7,0.5)); draw((3.7,0.7)--(3.7,1),EndArrow(TeXHead)); draw((3.7,0.3)--(3.7,0),EndArrow(TeXHead)); [/asy]

2

Let the length of the inner rectangle be $x$. Then the area of that rectangle is $x\cdot1 = x$. The second largest rectangle has dimensions of $x+2$ and $3$, making its area $3x+6$. The area of the second shaded area, therefore, is $3x+6-x = 2x+6$. The largest rectangle has dimensions of $x+4$ and $5$, making its area $5x + 20$. The area of the largest shaded region is the largest rectangle minus the second largest rectangle, which is $(5x+20) - (3x+6) = 2x + 14$. The problem states that $x, 2x+6, 2x+14$ is an arithmetic progression, meaning that the terms in the sequence increase by the same amount each term. Therefore, $(2x+6) - (x) = (2x+14) - (2x+6)\implies x+6 = 8\implies x =2\implies \fbox{2}$

AMC10 First Half

AMC10 A

HMMT

HMMT-Nov

thm

Nov

Let $z$ be a complex number and $k$ a positive integer such that $z^{k}$ is a positive real number other than 1. Let $f(n)$ denote the real part of the complex number $z^{n}$. Assume the parabola $p(n)=a n^{2}+b n+c$ intersects $f(n)$ four times, at $n=0,1,2,3$. Assuming the smallest possible value of $k$, find the largest possible value of $a$.

\frac{1}{3}

Let $r=|z|, \theta=\arg z$, and $C=\frac{\Re z}{|z|}=\cos \theta=\cos \frac{2 \pi j}{k}$ for some $j$ with $\operatorname{gcd}(j, k)=1$. The condition of the four consecutive points lying on a parabola is equivalent to having the finite difference \[ f(3)-3 f(2)+3 f(1)-f(0)=0 \] This implies \[ \begin{aligned} f(3)-f(0) & =3[f(2)-f(1)] \\ \Longleftrightarrow r^{3} \cos (3 \theta)-1 & =3\left(r^{2} \cos (2 \theta)-r \cos (\theta)\right) \\ \Longleftrightarrow r^{3}\left(4 C^{3}-3 C\right)-1 & =3\left(r^{2}\left(2 C^{2}-1\right)-r C\right) \end{aligned} \] Now we simply test the first few possible values of $k$. $k=1$ implies $C=1$, which gives $r^{3}-1=3\left(r^{2}-r\right) \Longrightarrow(r-1)^{3}=0 \Longrightarrow r=1$. This is not allowed since $r=1$ implies a periodic function. $k=2$ implies $C=-1$, which gives $-r^{3}-1=3 r^{2}+r \Longrightarrow(r+1)^{3}=0$, again not allowed since $r>0$. $k=3$ implies $C=-\frac{1}{2}$. This gives $r^{3}-1=\frac{-3}{2}\left(r^{2}-r\right) \Longrightarrow(r-1)\left(r+\frac{1}{2}\right)(r+2)=0$. These roots are either negative or 1 , again not allowed. $k=4$ implies $C=0$. This gives $-1=-3 r^{2} \Longrightarrow r= \pm \frac{1}{\sqrt{3}} \cdot r=\frac{1}{\sqrt{3}}$ is allowed, so this will generate our answer. Again by finite differences (or by any other method of interpolating with a quadratic), we get $2 a=$ $f(0)+f(2)-2 f(1)=\frac{2}{3}$, so $a=\frac{1}{3}$. $\fbox{\frac{1}{3}}$.

HMMT Nov Hard

HMMT-Nov Theme

AMC

AMC10

10B

N/A

Two concentric circles have radii $1$ and $2$. Two points on the outer circle are chosen independently and uniformly at random. What is the probability that the chord joining the two points intersects the inner circle?

\frac{1}{3}

Let the center of the two circles be $O$. Now pick an arbitrary point $A$ on the boundary of the circle with radius $2$. We want to find the range of possible places for the second point, $A'$, such that $AA'$ passes through the circle of radius $1$. To do this, first draw the tangents from $A$ to the circle of radius $1$. Let the intersection points of the tangents (when extended) with circle of radius $2$ be $B$ and $C$. Let $H$ be the foot of the altitude from $O$ to $\overline{BC}$. Then we have the following diagram. [asy] scale(200); pair A,O,B,C,H; A = (0,1); O = (0,0); B = (-.866,-.5); C = (.866,-.5); H = (0, -.5); draw(A--C--cycle); draw(A--O--cycle); draw(O--C--cycle); draw(O--H,dashed+linewidth(.7)); draw(A--B--cycle); draw(B--C--cycle); draw(O--B--cycle); dot("$A$",A,N); dot("$O$",O,NW); dot("$B$",B,W); dot("$C$",C,E); dot("$H$",H,S); label("$2$",O--(-.7,-.385),N); label("$1$",O--H,E); draw(circle(O,.5)); draw(circle(O,1)); [/asy] We want to find $\angle BOC$, as the range of desired points $A'$ is the set of points on minor arc $\overarc{BC}$. This is because $B$ and $C$ are part of the tangents, which "set the boundaries" for $A'$. Since $OH = 1$ and $OB = 2$ as shown in the diagram, $\triangle OHB$ is a $30-60-90$ triangle with $\angle BOH = 60^\circ$. Thus, $\angle BOC = 120^\circ$, and the probability $A'$ lies on the minor arc $\overarc{BC}$ is thus $\dfrac{120}{360} = \fbox{\frac{1}{3}}$.

AMC10 Second Half

AMC10 B

HMMT

HMMT-Feb

alg

Feb

Suppose $P(x)$ is a monic polynomial of degree 2023 such that \[ P(k)=k^{2023} P\left(1-\frac{1}{k}\right) \] for every positive integer $1 \leq k \leq 2023$. Then $P(-1)=\frac{a}{b}$, where $a$ and $b$ relatively prime integers. Compute the unique integer $0 \leq n<2027$ such that $b n-a$ is divisible by the prime 2027 .

406

Solution: Let $n=2023$. If $P(x)=x^{n}+a_{n-1} x^{n-1}+\cdots+a_{0}$, then let \[ R(x)=x^{n} P\left(1-\frac{1}{x}\right)=(x-1)^{n}+a_{n-1}(x-1)^{n} x+\cdots+a_{0} x^{n} \] Then, note that $Q(x)=P(x)-R(x)$ is a polynomial of degree at most $n$, and it has roots $1,2, \ldots, n$, so we have $Q(x)=k(x-1) \cdots(x-n)$ for some real constant $k$. Now we determine $P(x)$ in terms of $Q(x)$. If $g(x)=1-1 / x$, then $g(g(x))=\frac{1}{1-x}$ and $g(g(g(x)))=x$. Therefore, we have \[ \begin{aligned} P(x)-x^{n} P\left(1-\frac{1}{x}\right) & =Q(x) \\ P\left(1-\frac{1}{x}\right)-\left(1-\frac{1}{x}\right)^{n} P\left(\frac{1}{1-x}\right) & =Q\left(1-\frac{1}{x}\right) \\ P\left(\frac{1}{1-x}\right)-\left(\frac{1}{1-x}\right)^{n} P(x) & =Q\left(\frac{1}{1-x}\right) \end{aligned} \] Adding the first equation, $x^{n}$ times the second, and $(x-1)^{n}$ times the third yields \[ 2 P(x)=Q(x)+x^{n} Q\left(\frac{x-1}{x}\right)+(x-1)^{n} Q\left(\frac{1}{1-x}\right) \] so \[ \begin{aligned} P(x)=\frac{k}{2}((x-1)(x-2) \cdots(x-n)+(0 x-1)(-1 x & -1) \cdots(-(n-1) x-1) \\ & +(-1 x+0)(-2 x+1) \cdots(-n x+(n-1))) \end{aligned} \] Therefore, \[ P(-1)=\frac{k}{2}(-(n+1) !+0+(2 n+1) ! !) \] Also, since $P$ is monic, we know that \[ 1=\frac{k}{2}(1+0-n !) \] so \[ P(-1)=\frac{(2 n-1) ! !-(n+1) !}{1-n !} \] Modulo 2027, $(n+1) !=2026 ! /(2026 \cdot 2025) \equiv-1 /(-1 \cdot-2) \equiv-1 / 2$ and $n !=(n+1) ! / 2024 \equiv 1 / 6$. Also, $(2 n+1) ! ! \equiv 0$. So our answer is \[ \frac{1 / 2}{1-1 / 6}=\frac{3}{5} \equiv \frac{2030}{5}=406 \] $\fbox{406}$.

HMMT Feb Hard

HMMT-Feb Algebra

HMMT

HMMT-Feb

guts

Feb

Bob is coloring lattice points in the coordinate plane. Find the number of ways Bob can color five points in $\{(x, y) \mid 1 \leq x, y \leq 5\}$ blue such that the distance between any two blue points is not an integer.

80

We can see that no two blue points can have the same $x$ or $y$ coordinate. The blue points then must make a permutation of $1,2,3,4,5$ that avoid the pattern of 3-4-5 triangles. It is not hard to use complementary counting to get the answer from here. There are 8 possible pairs of points that are a distance of 5 apart while not being in the same row or column (i.e. a pair that is in the 3-4-5 position). If such a pair of points is included in the choice of five points, then there are $3 !=6$ possibilities for the remaining three points, yielding $8 \times 6=48$ configurations that have violations. However, we now need to consider overcounting. The only way to have more than one violation in one configuration is to have a corner point and then two points adjacent to the opposite corner, e.g. $(1,1),(4,5),(5,4)$. In each such case, there are exactly $2 !=2$ possibilities for the other two points, and there are exactly two violations so there are a total of $2 \times 4=8$ configurations that are double-counted. Therefore, there are $48-8=40$ permutations that violate the no-integer-condition, leaving $120-40=$ 80 good configurations. $\fbox{80}$.

HMMT Feb Guts

HMMT-Feb Guts

AMC

AMC10

10B

N/A

What is the sum of all the solutions of $x = \left|2x-|60-2x|\right|$?

92

We evaluate this in cases: Case 1 $x<30$ When $x<30$ we are going to have $60-2x>0$. When $x>0$ we are going to have $|x|>0\implies x>0$ and when $-x>0$ we are going to have $|x|>0\implies -x>0$. Therefore we have $x=|2x-(60-2x)|$. $x=|2x-60+2x|\implies x=|4x-60|$ Subcase 1 $30>x>15$ When $30>x>15$ we are going to have $4x-60>0$. When this happens, we can express $|4x-60|$ as $4x-60$. Therefore we get $x=4x-60\implies -3x=-60\implies x=20$. We check if $x=20$ is in the domain of the numbers that we put into this subcase, and it is, since $30>20>15$. Therefore $20$ is one possible solution. Subcase 2 $x<15$ When $x<15$ we are going to have $4x-60<0$, therefore $|4x-60|$ can be expressed in the form $60-4x$. We have the equation $x=60-4x\implies 5x=60\implies x=12$. Since $12$ is less than $15$, $12$ is another possible solution. $x=|2x-|60-2x||$ Case 2 : $x>30$ When $x>30$, $60-2x<0$. When $x<0$ we can express this in the form $-x$. Therefore we have $-(60-2x)=2x-60$. This makes sure that this is positive, since we just took the negative of a negative to get a positive. Therefore we have $x=|2x-(2x-60)|$ $x=|2x-2x+60|$ $x=|60|$ $x=60$ We have now evaluated all the cases, and found the solution to be $\{60,12,20\}$ which have a sum of $\fbox{92}$

AMC10 Second Half

AMC10 B

AMC

AMC8

8

N/A

How many 4-digit positive integers have four different digits, where the leading digit is not zero, the integer is a multiple of 5, and 5 is the largest digit?

84

We can separate this into two cases. If an integer is a multiple of $5,$ the last digit must be either $0$ or $5.$ Case 1: The last digit is $5.$ The leading digit can be $1,2,3,$ or $4.$ Because the second digit can be $0$ but not the leading digit, there are also $4$ choices. The third digit cannot be the leading digit or the second digit, so there are $3$ choices. The number of integers is this case is $4\cdot4\cdot3\cdot1=48.$ Case 2: The last digit is $0.$ Because $5$ is the largest digit, one of the remaining three digits must be $5.$ There are $3$ ways to choose which digit should be $5.$ The remaining digits can be $1,2,3,$ or $4,$ but since they have to be different there are $4\cdot3$ ways to choose. The number of integers in this case is $1\cdot3\cdot4\cdot3=36.$ Therefore, the answer is $48+36=\fbox{84}$.

AMC8 Second Half

AMC8

AMC

AMC12

12B

N/A

In $\triangle ABC$, $AB=13$, $AC=5$, and $BC=12$. Points $M$ and $N$ lie on $AC$ and $BC$, respectively, with $CM=CN=4$. Points $J$ and $K$ are on $AB$ so that $MJ$ and $NK$ are perpendicular to $AB$. What is the area of pentagon $CMJKN$? [asy] unitsize(0.5cm); defaultpen(0.8); pair C=(0,0), A=(0,5), B=(12,0), M=(0,4), N=(4,0); pair J=intersectionpoint(A--B, M--(M+rotate(90)*(B-A)) ); pair K=intersectionpoint(A--B, N--(N+rotate(90)*(B-A)) ); draw( A--B--C--cycle ); draw( M--J ); draw( N--K ); label("$A$",A,NW); label("$B$",B,SE); label("$C$",C,SW); label("$M$",M,SW); label("$N$",N,S); label("$J$",J,NE); label("$K$",K,NE); [/asy]

\frac{240}{13}

The triangle $ABC$ is clearly a right triangle, its area is $\frac{5\cdot 12}2 = 30$. If we knew the areas of triangles $AMJ$ and $BNK$, we could subtract them to get the area of the pentagon. Draw the height $CL$ from $C$ onto $AB$. As $AB=13$ and the area is $30$, we get $CL=\frac{60}{13}$. The situation is shown in the picture below: [asy] unitsize(0.5cm); defaultpen(0.8); pair C=(0,0), A=(0,5), B=(12,0), M=(0,4), N=(4,0); pair J=intersectionpoint(A--B, M--(M+rotate(90)*(B-A)) ); pair K=intersectionpoint(A--B, N--(N+rotate(90)*(B-A)) ); pair L=intersectionpoint(A--B, C--(C+rotate(90)*(B-A)) ); draw( A--B--C--cycle ); draw( M--J ); draw( N--K ); draw( C--L, dashed ); label("$A$",A,NW); label("$B$",B,SE); label("$C$",C,SW); label("$M$",M,SW); label("$N$",N,S); label("$J$",J,NE); label("$K$",K,NE); label("$L$",L,NE); [/asy] Now note that the triangles $ABC$, $AMJ$, $ACL$, $CBL$ and $NBK$ all have the same angles and therefore they are similar. We already know some of their sides, and we will use this information to compute their areas. Note that if two polygons are similar with ratio $k$, their areas have ratio $k^2$. We will use this fact repeatedly. Below we will use $[XYZ]$ to denote the area of the triangle $XYZ$. We have $\frac{CL}{BC} = \frac{60/13}{12} = \frac 5{13}$, hence $[ACL] = \frac{ 25[ABC] }{169} = \frac{750}{169}$. Also, $\frac{CL}{AC} = \frac{60/13}5 = \frac{12}{13}$, hence $[CBL] = \frac{ 144[ABC] }{169} = \frac{4320}{169}$. Now for the smaller triangles: We know that $\frac{AM}{AC}=\frac 15$, hence $[AMJ] = \frac{[ACL]}{25} = \frac{30}{169}$. Similarly, $\frac{BN}{BC}=\frac 8{12} = \frac 23$, hence $[NBK] = \frac{4[CBL]}9 = \frac{1920}{169}$. Finally, the area of the pentagon is $30 - \frac{30}{169} - \frac{1920}{169} = \fbox{\frac{240}{13}}$.

AMC12 Second Half

AMC12 B

HMMT

HMMT-Feb

guts

Feb

Let $A B C$ be a triangle with $A B=6, A C=7, B C=8$. Let $I$ be the incenter of $A B C$. Points $Z$ and $Y$ lie on the interior of segments $A B$ and $A C$ respectively such that $Y Z$ is tangent to the incircle. Given point $P$ such that \[ \angle Z P C=\angle Y P B=90^{\circ} \] find the length of $I P$.

\frac{\sqrt{30}}{2}

Solution 1. Let $P U, P V$ tangent from $P$ to the incircle. We will invoke the dual of the Desargues Involution Theorem, which states the following: Given a point $P$ in the plane and four lines $\ell_{1}, \ell_{2}, \ell_{3}, \ell_{4}$, consider the set of conics tangent to all four lines. Then we define a function on the pencil of lines through $P$ by mapping one tangent from $P$ to each conic to the other. This map is well defined and is a projective involution, and in particular maps $P A \rightarrow P D, P B \rightarrow P E, P C \rightarrow P F$, where $A B C D E F$ is the complete quadrilateral given by the pairwise intersections of $\ell_{1}, \ell_{2}, \ell_{3}, \ell_{4}$. An overview of the projective background behind the (Dual) Desargues Involution Theorem can be found here: \href{https://www.scribd.com/document/384321704/Desargues-Involution-Theorem}{https://www.scribd.com/document/384321704/Desargues-Involution-Theorem}, and a proof can be found at \href{https://www2.washjeff.edu/users/mwoltermann/Dorrie/63.pdf}{https://www2.washjeff.edu/users/mwoltermann/Dorrie/63.pdf}. Now, we apply this to the point $P$ and the lines $A B, A C, B C, Y Z$, to get that the pairs \[ (P U, P V),(P Y, P B),(P Z, P C) \] are swapped by some involution. But we know that the involution on lines through $P$ which rotates by $90^{\circ}$ swaps the latter two pairs, thus it must also swap the first one and $\angle U P V=90$. It follows by equal tangents that $I U P V$ is a square, thus $I P=r \sqrt{2}$ where $r$ is the inradius of $A B C$. Since $r=\frac{2 K}{a+b+c}=\frac{21 \sqrt{15} / 2}{21}=\frac{\sqrt{15}}{2}$, we have $I P=\frac{\sqrt{30}}{2}$. Solution 2. Let $H$ be the orthocenter of $A B C$. Lemma. $H I^{2}=2 r^{2}-4 R^{2} \cos (A) \cos (B) \cos (C)$, where $r$ is the inradius and $R$ is the circumradius. Proof. This follows from barycentric coordinates or the general result that for a point $X$ in the plane, \[ a X A^{2}+b X B^{2}+c X C^{2}=(a+b+c) X I^{2}+a A I^{2}+b B I^{2}+c C I^{2} \] which itself is a fact about vectors that follows from barycentric coordinates. This can also be computed directly using trigonometry. Let $E=B H \cap A C, F=C H \cap A B$, then note that $B, P, E, Y$ are concyclic on the circle of diameter $B Y$, and $C, P, F, Z$ are concyclic on the circle of diameter $C Z$. Let $Q$ be the second intersection of these circles. Since $B C Y Z$ is a tangential quadrilateral, the midpoints of $B Y$ and $C Z$ are collinear with $I$ (this is known as Newton's theorem), which implies that $I P=I Q$ by symmetry. Note that as $B H \cdot H E=C H \cdot H F, H$ lies on the radical axis of the two circles, which is $P Q$. Thus, if $I P=I Q=x$, $B H \cdot H E$ is the power of $H$ with respect to the circle centered at $I$ with radius $x$, which implies $B H \cdot H E=x^{2}-H I^{2}$. As with the first solution, we claim that $x=r \sqrt{2}$, which by the lemma is equivalent to $B H \cdot H E=$ $4 R^{2} \cos (A) \cos (B) \cos (C)$. Then note that \[ B H \cdot H E=B H \cdot C H \cos (A)=(2 R \cos (B))(2 R \cos (C)) \cos (A) \] so our claim holds and we finish as with the first solution. Note. Under the assumption that the problem is well-posed (the answer does not depend on the choice of $Y, Z$, or $P$ ), then here is an alternative method to obtain $I P=r \sqrt{2}$ by making convenient choices. Let $U$ be the point where $Y Z$ is tangent to the incircle, and choose $U$ so that $I U \| B C$ (and therefore $Y Z \perp B C)$. Note that $Y Z \cap B C$ is a valid choice for $P$, so assume that $P$ is the foot from $U$ to $B C$. If $D$ is the point where $B C$ is tangent to the incircle, then $I U P D$ is a square so $I P=r \sqrt{2}$. (This disregards the condition that $Y$ and $Z$ are in the interior of segments $A C$ and $A B$, but there is no reason to expect that this condition is important.) $\fbox{\frac{\sqrt{30}}{2}}$.

HMMT Feb Guts

HMMT-Feb Guts

AIME

AIME

II

N/A

In $\triangle RED$, $\measuredangle DRE=75^{\circ}$ and $\measuredangle RED=45^{\circ}$. $RD=1$. Let $M$ be the midpoint of segment $\overline{RD}$. Point $C$ lies on side $\overline{ED}$ such that $\overline{RC}\perp\overline{EM}$. Extend segment $\overline{DE}$ through $E$ to point $A$ such that $CA=AR$. Then $AE=\frac{a-\sqrt{b}}{c}$, where $a$ and $c$ are relatively prime positive integers, and $b$ is a positive integer. Find $a+b+c$.

56

Let $P$ be the foot of the perpendicular from $A$ to $\overline{CR}$, so $\overline{AP}\parallel\overline{EM}$. Since triangle $ARC$ is isosceles, $P$ is the midpoint of $\overline{CR}$, and $\overline{PM}\parallel\overline{CD}$. Thus, $APME$ is a parallelogram and $AE = PM = \frac{CD}{2}$. We can then use coordinates. Let $O$ be the foot of altitude $RO$ and set $O$ as the origin. Now we notice special right triangles! In particular, $DO = \frac{1}{2}$ and $EO = RO = \frac{\sqrt{3}}{2}$, so $D\left(\frac{1}{2}, 0\right)$, $E\left(-\frac{\sqrt{3}}{2}, 0\right)$, and $R\left(0, \frac{\sqrt{3}}{2}\right).$ $M =$ midpoint$(D, R) = \left(\frac{1}{4}, \frac{\sqrt{3}}{4}\right)$ and the slope of $ME = \frac{\frac{\sqrt{3}}{4}}{\frac{1}{4} + \frac{\sqrt{3}}{2}} = \frac{\sqrt{3}}{1 + 2\sqrt{3}}$, so the slope of $RC = -\frac{1 + 2\sqrt{3}}{\sqrt{3}}.$ Instead of finding the equation of the line, we use the definition of slope: for every $CO = x$ to the left, we go $\frac{x(1 + 2\sqrt{3})}{\sqrt{3}} = \frac{\sqrt{3}}{2}$ up. Thus, $x = \frac{\frac{3}{2}}{1 + 2\sqrt{3}} = \frac{3}{4\sqrt{3} + 2} = \frac{3(4\sqrt{3} - 2)}{44} = \frac{6\sqrt{3} - 3}{22}.$ $DC = \frac{1}{2} - x = \frac{1}{2} - \frac{6\sqrt{3} - 3}{22} = \frac{14 - 6\sqrt{3}}{22}$, and $AE = \frac{7 - \sqrt{27}}{22}$, so the answer is $\fbox{56}$. [asy] unitsize(8cm); pair a, o, d, r, e, m, cm, c,p; o =(0,0); d = (0.5, 0); r = (0,sqrt(3)/2); e = (-sqrt(3)/2,0); m = midpoint(d--r); draw(e--m); cm = foot(r, e, m); draw(L(r, cm,1, 1)); c = IP(L(r, cm, 1, 1), e--d); clip(r--d--e--cycle); draw(r--d--e--cycle); draw(rightanglemark(e, cm, c, 1.5)); a = -(4sqrt(3)+9)/11+0.5; dot(a); draw(a--r, dashed); draw(a--c, dashed); pair[] PPAP = {a, o, d, r, e, m, c}; for(int i = 0; i<7; ++i) { dot(PPAP[i]); } label("$A$", a, W); label("$E$", e, SW); label("$C$", c, S); label("$O$", o, S); label("$D$", d, SE); label("$M$", m, NE); label("$R$", r, N); p = foot(a, r, c); label("$P$", p, NE); draw(p--m, dashed); draw(a--p, dashed); dot(p); [/asy]

Hard AIME Problems

AIME

AMC

AMC12

12B

N/A

The graph of $2x^2 + xy + 3y^2 - 11x - 20y + 40 = 0$ is an ellipse in the first quadrant of the $xy$-plane. Let $a$ and $b$ be the maximum and minimum values of $\frac yx$ over all points $(x,y)$ on the ellipse. What is the value of $a+b$?

\frac 72

$\frac yx$ represents the slope of a line passing through the origin. It follows that since a line $y = mx$ intersects the ellipse at either $0, 1,$ or $2$ points, the minimum and maximum are given when the line $y = mx$ is a tangent, with only one point of intersection. Substituting, \[2x^2 + x(mx) + 3(mx)^2 - 11x - 20(mx) + 40 = 0\] Rearranging by the degree of $x$, \[(3m^2 + m + 2)x^2 - (20m + 11)x + 40 = 0\] Since the line $y=mx$ is tangent to the ellipse, we want the discriminant, \[(20m+11)^2 - 4\cdot 40 \cdot (3m^2 + m + 2) = -80m^2 + 280m - 199\] to be equal to $0$. We want $a+b$, which is the sum of the roots of the above quadratic. By Vieta’s formulas, that is $\frac{280}{80} = \frac{7}{2} $. $\fbox{\frac 72}$.

AMC12 Final Problems

AMC12 B

AMC

AMC10

10A

N/A

Ana and Bonita were born on the same date in different years, $n$ years apart. Last year Ana was $5$ times as old as Bonita. This year Ana's age is the square of Bonita's age. What is $n?$

12

Let $A$ be the age of Ana and $B$ be the age of Bonita. Then, \[A-1 = 5(B-1)\] and \[A = B^2.\] Substituting the second equation into the first gives us \[B^2-1 = 5(B-1).\] By using difference of squares and dividing, $B=4.$ Moreover, $A=B^2=16.$ The answer is $16-4 = 12 \implies \fbox{12}$

AMC10 First Half

AMC10 A

HMMT

HMMT-Feb

team

Feb

Thaddeus is given a $2013 \times 2013$ array of integers each between 1 and 2013 , inclusive. He is allowed two operations: Choose a row, and subtract 1 from each entry. Chooses a column, and add 1 to each entry. He would like to get an array where all integers are divisible by 2013. On how many arrays is this possible?

2013^{4025}

We claim that the set of grids on which it is possible to obtain an array of all zeroes (mod 2013) is indexed by ordered 4025-tuples of residues ( $\bmod 2013)$, corresponding to the starting entries in the first row and first column of the grid, giving the answer of $2013^{4025}$. To do this, we show that given after fixing all of the entries in the first row and column, there is a unique starting grid which can become an array of all zeroes after applying the appropriate operations. Let $a_{i, j}$ be the entry in the $i$-th row and the $j$-th column. Suppose there is a sequence of operations giving all zeroes in the array; let $r_{i}$ be the number of times we operate on row $i$, and let $c_{j}$ be the number of times we operate on column $j$. It is enough to take all of these values to be residues modulo 2013. Clearly, $a_{i, j}+r_{i}+c_{j} \equiv 0(\bmod 2013)$ for each $i, j$. In particular, $r_{1}+c_{1} \equiv a_{1,1}$. Now, for each $i, j$, we have \[ \begin{aligned} a_{i, j} & \equiv-r_{i}-c_{j} \\ & \equiv\left(a_{i, 1}+c_{1}\right)+\left(a_{1, j}+r_{1}\right) \\ & \equiv a_{i, 1}+a_{1, j}-a_{1,1} \end{aligned} \] which is fixed. Thus, there rest of the entries in the grid are forced. Conversely, if we set $a_{i, j}$ to be the appropriate representative of the residue class of $a_{i, 1}+a_{1, j}-a_{1,1}$ modulo 2013, we may take $r_{i} \equiv-a_{i, 1}(\bmod 2013)$, and $c_{j} \equiv a_{1,1}-a_{1, j}(\bmod 2013)$ for each $i, j$. It is clear that $a_{i, j}+r_{i}+c_{j} \equiv 0(\bmod 2013)$ for each $i, j$, so we're done. $\fbox{2013^{4025}}$.

HMMT Feb Team

HMMT-Feb Team

AIME

AIME

I

N/A

Kathy has $5$ red cards and $5$ green cards. She shuffles the $10$ cards and lays out $5$ of the cards in a row in a random order. She will be happy if and only if all the red cards laid out are adjacent and all the green cards laid out are adjacent. For example, card orders RRGGG, GGGGR, or RRRRR will make Kathy happy, but RRRGR will not. The probability that Kathy will be happy is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

157

We have $2+4\cdot 2$ cases total. The two are all red and all green. Then, you have 4 of one, 1 of other. 3 of one, 2 of other. 2 of one, 3 of other. 1 of one, 4 of other. Then flip the order, so times two. Obviously the denominator is $10\cdot 9\cdot 8\cdot 7\cdot 6$, since we are choosing a card without replacement. Then, we have for the numerator for the two of all red and green: \[5\cdot 4\cdot 3\cdot 2\cdot 1.\] For the 4 and 1, we have: \[5\cdot 4\cdot 3\cdot 2\cdot 5.\] For the 3 and 2, we have: \[5\cdot 4\cdot 3\cdot 5\cdot 4.\] For the 2 and 3, we have: \[5\cdot 4\cdot 5\cdot 4\cdot 3.\] For the 1 and 4, we have: \[5\cdot 5\cdot 4\cdot 3\cdot 2.\] Adding up and remembering to double all of them, since they can be reversed and the 5's can be red or green, we get, after simplifying: $\dfrac{31}{126}$ Thus the answer is $31 + 126 = \fbox{157}$. -gorefeebuddie

Easy AIME Problems

AIME

HMMT

HMMT-Nov

guts

Nov

The length of a rectangle is three times its width. Given that its perimeter and area are both numerically equal to $k>0$, find $k$.

\frac{64}{3}

Let $a$ be the width of the rectangle. Then the length of the rectangle is $3 a$, so the perimeter is $2(a+3 a)=8 a$, and the area is $3 a^{2}$. Since the length is numerically equal to the width, we know that \[ 8 a=3 a^{2}=k \] Because $k>0$, the rectangle is non-degenerate. It follows that $8=3 a$, so $a=\frac{8}{3}$. Therefore, $k=\frac{64}{3}$. $\fbox{\frac{64}{3}}$.

HMMT Nov Guts

HMMT-Nov Guts

AMC

AMC8

8

N/A

Each principal of Lincoln High School serves exactly one $3$-year term. What is the maximum number of principals this school could have during an $8$-year period?

4

If the first year of the $8$-year period was the final year of a principal's term, then in the next six years two more principals would serve, and the last year of the period would be the first year of the fourth principal's term. Therefore, the maximum number of principals who can serve during an $8$-year period is $4$, so the answer is $\fbox{4}$ if the terms are divided $1\ |\ 2\ 3\ 4\ |\ 5\ 6\ 7\ |\ 8$

AMC8 First Half

AMC8

HMMT

HMMT-Nov

guts

Nov

Let $M$ denote the number of positive integers which divide 2014!, and let $N$ be the integer closest to $\ln (M)$. Estimate the value of $N$. If your answer is a positive integer $A$, your score on this problem will be the larger of 0 and $\left\lfloor 20-\frac{1}{8}|A-N|\right\rfloor$. Otherwise, your score will be zero.

439

Combining Legendre's Formula and the standard prime approximations, the answer is \[ \prod_{p}\left(1+\frac{2014-s_{p}(2014)}{p-1}\right) \] where $s_{p}(n)$ denotes the sum of the base $p$-digits of $n$. Estimate $\ln 1000 \approx 8$, and $\ln 2014 \approx 9$. Using the Prime Number Theorem or otherwise, one might estimate about 150 primes less than 1007 and 100 primes between 1008 and 2014. Each prime between 1008 and 2014 contributes exactly $\ln 2$. For the other 150 primes we estimate $\ln 2014 / p$ as their contribution, which gives $\sum_{p<1000}(\ln 2014-\ln p)$. Estimating the average $\ln p$ for $p<1000$ to be $\ln 1000-1 \approx 7$ (i.e. an average prime less than 1000 might be around $1000 / e$ ), this becomes $150 \cdot 2=300$. So these wildly vague estimates give $300+150 \ln 2 \approx 400$, which is not far from the actual answer. The following program in Common Lisp then gives the precise answer of 438.50943. Guts Round $\fbox{439}$.

HMMT Nov Guts

HMMT-Nov Guts

AIME

AIME

I

N/A

Let $x,$ $y,$ and $z$ be positive real numbers satisfying the system of equations: \begin{align} \sqrt{2x-xy} + \sqrt{2y-xy} &= 1 \\ \sqrt{2y-yz} + \sqrt{2z-yz} &= \sqrt2 \\ \sqrt{2z-zx} + \sqrt{2x-zx} &= \sqrt3. \end{align} Then $\left[ (1-x)(1-y)(1-z) \right]^2$ can be written as $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$

33

First, let define a triangle with side lengths $\sqrt{2x}$, $\sqrt{2z}$, and $l$, with altitude from $l$'s equal to $\sqrt{xz}$. $l = \sqrt{2x - xz} + \sqrt{2z - xz}$, the left side of one equation in the problem. Let $\theta$ be angle opposite the side with length $\sqrt{2x}$. Then the altitude has length $\sqrt{2z} \cdot \sin(\theta) = \sqrt{xz}$ and thus $\sin(\theta) = \sqrt{\frac{x}{2}}$, so $x=2\sin^2(\theta)$ and the side length $\sqrt{2x}$ is equal to $2\sin(\theta)$. We can symmetrically apply this to the two other equations/triangles. By law of sines, we have $\frac{2\sin(\theta)}{\sin(\theta)} = 2R$, with $R=1$ as the circumradius, same for all 3 triangles. The circumcircle's central angle to a side is $2 \arcsin(l/2)$, so the 3 triangles' $l=1, \sqrt{2}, \sqrt{3}$, have angles $120^{\circ}, 90^{\circ}, 60^{\circ}$, respectively. This means that by half angle arcs, we see that we have in some order, $x=2\sin^2(\alpha)$, $y=2\sin^2(\beta)$, and $z=2\sin^2(\gamma)$ (not necessarily this order, but here it does not matter due to symmetry), satisfying that $\alpha+\beta=180^{\circ}-\frac{120^{\circ}}{2}$, $\beta+\gamma=180^{\circ}-\frac{90^{\circ}}{2}$, and $\gamma+\alpha=180^{\circ}-\frac{60^{\circ}}{2}$. Solving, we get $\alpha=\frac{135^{\circ}}{2}$, $\beta=\frac{105^{\circ}}{2}$, and $\gamma=\frac{165^{\circ}}{2}$. We notice that \[[(1-x)(1-y)(1-z)]^2=[\sin(2\alpha)\sin(2\beta)\sin(2\gamma)]^2=[\sin(135^{\circ})\sin(105^{\circ})\sin(165^{\circ})]^2\] \[=\left(\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{6}-\sqrt{2}}{4} \cdot \frac{\sqrt{6}+\sqrt{2}}{4}\right)^2 = \left(\frac{\sqrt{2}}{8}\right)^2=\frac{1}{32} \to \fbox{33}. \blacksquare\] - kevinmathz

Very Hard AIME Problems

AIME

HMMT

HMMT-Nov

gen

Nov

Evaluate the sum: \[ 11^{2}-1^{2}+12^{2}-2^{2}+13^{2}-3^{2}+\ldots+20^{2}-10^{2} \]

2100

This sum can be written as $\sum_{a=1}^{10}(a+10)^{2}-a^{2}=\sum_{a=1}^{10} 10(2 a+10)=10 * 10 *$ $11+10 * 10 * 10=2100$. $\fbox{2100}$.

HMMT Nov Easy

HMMT-Nov General

HMMT

HMMT-Nov

guts

Nov

If $f(x)=x /(x+1)$, what is $f(f(f(f(2009))))$ ?

\frac{2009}{8037}

$f(f(x))=\frac{(x /(x+1))}{(x /(x+1))+1}=x / 2 x+1, f(f(f(f(x))))=x / 4 x+1=\frac{2009}{8037}$ $\fbox{\frac{2009}{8037}}$.

HMMT Nov Guts

HMMT-Nov Guts

AMC

AMC12

12A

N/A

Triangle $ABC$ is an isosceles right triangle with $AB=AC=3$. Let $M$ be the midpoint of hypotenuse $\overline{BC}$. Points $I$ and $E$ lie on sides $\overline{AC}$ and $\overline{AB}$, respectively, so that $AI>AE$ and $AIME$ is a cyclic quadrilateral. Given that triangle $EMI$ has area $2$, the length $CI$ can be written as $\frac{a-\sqrt{b}}{c}$, where $a$, $b$, and $c$ are positive integers and $b$ is not divisible by the square of any prime. What is the value of $a+b+c$?

12

Observe that $\triangle{EMI}$ is isosceles right ($M$ is the midpoint of diameter arc $EI$ since $m\angle MEI = m\angle MAI = 45^\circ$), so $MI=2,MC=\frac{3}{\sqrt{2}}$. With $\angle{MCI}=45^\circ$, we can use Law of Cosines to determine that $CI=\frac{3\pm\sqrt{7}}{2}$. The same calculations hold for $BE$ also, and since $CI<BE$, we deduce that $CI$ is the smaller root, giving the answer of $\fbox{12}$.

AMC12 Second Half

AMC12 A

AIME

AIME

I

N/A

The domain of the function $f(x) = \arcsin(\log_{m}(nx))$ is a closed interval of length $\frac{1}{2013}$ , where $m$ and $n$ are positive integers and $m>1$. Find the remainder when the smallest possible sum $m+n$ is divided by 1000.

371

We know that the domain of $\text{arcsin}$ is $[-1, 1]$, so $-1 \le \log_m nx \le 1$. Now we can apply the definition of logarithms: \[m^{-1} = \frac1m \le nx \le m\] \[\implies \frac{1}{mn} \le x \le \frac{m}{n}\] Since the domain of $f(x)$ has length $\frac{1}{2013}$, we have that \[\frac{m}{n} - \frac{1}{mn} = \frac{1}{2013}\] \[\implies \frac{m^2 - 1}{mn} = \frac{1}{2013}\] A larger value of $m$ will also result in a larger value of $n$ since $\frac{m^2 - 1}{mn} \approx \frac{m^2}{mn}=\frac{m}{n}$ meaning $m$ and $n$ increase about linearly for large $m$ and $n$. So we want to find the smallest value of $m$ that also results in an integer value of $n$. The problem states that $m > 1$. Thus, first we try $m = 2$: \[\frac{3}{2n} = \frac{1}{2013} \implies 2n = 3 \cdot 2013 \implies n \notin \mathbb{Z}\] Now, we try $m=3$: \[\frac{8}{3n} = \frac{1}{2013} \implies 3n = 8 \cdot 2013 \implies n = 8 \cdot 671 = 5368\] Since $m=3$ is the smallest value of $m$ that results in an integral $n$ value, we have minimized $m+n$, which is $5368 + 3 = 5371 \equiv \fbox{371} \pmod{1000}$.

Intermediate AIME Problems

AIME

HMMT

HMMT-Feb

team

Feb

Does there exist an irrational number $\alpha>1$ such that \[ \left\lfloor\alpha^{n}\right\rfloor \equiv 0 \quad(\bmod 2017) \] for all integers $n \geq 1$ ?

Yes

Let $\alpha>1$ and $0<\beta<1$ be the roots of $x^{2}-4035 x+2017$. Then note that $\left\lfloor\alpha^{n}\right\rfloor=\alpha^{n}+\beta^{n}-1$. Let $x_{n}=\alpha^{n}+\beta^{n}$ for all nonnegative integers $n$. It's easy to verify that $x_{n}=4035 x_{n-1}-2017 x_{n-2} \equiv x_{n-1}$ $(\bmod 2017)$ so since $x_{1}=4035 \equiv 1(\bmod 2017)$ we have that $x_{n} \equiv 1(\bmod 2017)$ for all $n$. Thus $\alpha$ satisfies the problem. $\fbox{Yes}$.

HMMT Feb Team

HMMT-Feb Team

AIME

AIME

I

N/A

The nine delegates to the Economic Cooperation Conference include $2$ officials from Mexico, $3$ officials from Canada, and $4$ officials from the United States. During the opening session, three of the delegates fall asleep. Assuming that the three sleepers were determined randomly, the probability that exactly two of the sleepers are from the same country is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

139

One of the best ways to solve this problem is to use PIE, or the Principle of Inclusion and Exclusion. To start off, we know that the denominator, or the total ways to pick $3$ officials, is $\binom{9}{3} = 84$. Now, we find the number of ways that at least $2$ officials are from the same country and then subtract the number of ways all $3$ officials are from the same country. To start with at least $2$ officials, we know: There are $7$ different ways to pick $3$ delegates such that $2$ are from Mexico, simply because there are $9-2=7$ "extra" delegates to choose to be the third sleeper once both from Mexico are sleeping. There are $3\times7=21$ ways to pick from Canada, as we choose $2$ of the $3$ Canadians ($\binom{3}{2} = 3$) and then there are $7$ other options for the third sleeper. Lastly, there are $6\times7=42$ ways to choose for the United States. We can choose two American officials with $\binom{4}{2} = 6$. Then, there are $7$ options for the third sleeper. Now, we want to find the number of ways to have three sleepers from the same country. There are no ways for the $3$ sleepers to be from Mexico because there are only $2$ Mexican officials. Hence, we get $0$ ways. There is only $1$ way to pick all $3$ from Canada because there are exactly $3$ Canadian officials. We now consider the number of times we originally counted this, which after inspection, is $3$, so we have $1 * 3 = 3$. Lastly, there are $4$ ways to choose all $3$ officials from the United States ($\binom{4}{3} = 4$). Once again, we counted this $3$ times, so we have $4*3 = 12$. Thus, the fraction is $\frac{7+21+42-0-3-12}{84} = \frac{55}{84}$, and our answer is $55+84=\fbox{139}$. Note: Similar to Solution 2, we could also have grouped each of the countries in terms of the ways we have for at least $2$ officials to sleep and for all $3$ to sleep. Mexico would have $7 - 0 = 7$ ways for exactly 2 officials to be there, Canada with $21 - 3 = 18$ ways, and the United States with $42 - 12 = 30$ ways. Solution by: armang32324

Easy AIME Problems

AIME

HMMT

HMMT-Nov

team

Nov

Let $A, B, C, D$ be points chosen on a circle, in that order. Line $B D$ is reflected over lines $A B$ and $D A$ to obtain lines $\ell_{1}$ and $\ell_{2}$ respectively. If lines $\ell_{1}, \ell_{2}$, and $A C$ meet at a common point and if $A B=4, B C=3, C D=2$, compute the length $D A$.

\sqrt{21}

Let the common point be $E$. Then since lines $B E$ and $B D$ are symmetric about line $B A, B A$ is an exterior bisector of $\angle D B E$, and similarly $D A$ is also an exterior bisector of $\angle B D E$. Therefore $A$ is the $E$-excenter of triangle $B D E$ and thus lie on the interior bisector of $\angle B E D$. Since $C$ lies on line $A E$, and by the fact that $A, B, C, D$ are concyclic, we get that $\angle A B C+\angle A D C=180^{\circ}$, which implies $\angle D B C+\angle B D C=\frac{1}{2}(\angle D B E+\angle B D E)$, so $C$ is the incenter of triangle $B D E$. Thus $\angle A B C=\angle C D A=\frac{\pi}{2}$, and thus $D A^{2}=A C^{2}-C D^{2}=A B^{2}+B C^{2}-C D^{2}=3^{2}+4^{2}-2^{2}=21$. The length of $D A$ is then $\sqrt{21}$. $\fbox{\sqrt{21}}$.

HMMT Nov Team

HMMT-Nov Team

AIME

AIME

II

N/A

In $\triangle{ABC}, AB=10, \angle{A}=30^\circ$ , and $\angle{C=45^\circ}$. Let $H, D,$ and $M$ be points on the line $BC$ such that $AH\perp{BC}$, $\angle{BAD}=\angle{CAD}$, and $BM=CM$. Point $N$ is the midpoint of the segment $HM$, and point $P$ is on ray $AD$ such that $PN\perp{BC}$. Then $AP^2=\dfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

77

Let us just drop the perpendicular from $B$ to $AC$ and label the point of intersection $O$. We will use this point later in the problem. As we can see, $M$ is the midpoint of $BC$ and $N$ is the midpoint of $HM$ $AHC$ is a $45-45-90$ triangle, so $\angle{HAB}=15^\circ$. $AHD$ is $30-60-90$ triangle. $AH$ and $PN$ are parallel lines so $PND$ is $30-60-90$ triangle also. Then if we use those informations we get $AD=2HD$ and $PD=2ND$ and $AP=AD-PD=2HD-2ND=2HN$ or $AP=2HN=HM$ Now we know that $HM=AP$, we can find for $HM$ which is simpler to find. We can use point $B$ to split it up as $HM=HB+BM$, We can chase those lengths and we would get $AB=10$, so $OB=5$, so $BC=5\sqrt{2}$, so $BM=\dfrac{1}{2} \cdot BC=\dfrac{5\sqrt{2}}{2}$ We can also use Law of Sines: \[\frac{BC}{AB}=\frac{\sin\angle A}{\sin\angle C}\] \[\frac{BC}{10}=\frac{\frac{1}{2}}{\frac{\sqrt{2}}{2}}\implies BC=5\sqrt{2}\] Then using right triangle $AHB$, we have $HB=10 \sin 15^\circ$ So $HB=10 \sin 15^\circ=\dfrac{5(\sqrt{6}-\sqrt{2})}{2}$. And we know that $AP = HM = HB + BM = \frac{5(\sqrt6-\sqrt2)}{2} + \frac{5\sqrt2}{2} = \frac{5\sqrt6}{2}$. Finally if we calculate $(AP)^2$. $(AP)^2=\dfrac{150}{4}=\dfrac{75}{2}$. So our final answer is $75+2=77$. $m+n=\fbox{77}$ -Gamjawon -edited by srisainandan6 to clarify and correct a small mistake

Very Hard AIME Problems

AIME

HMMT

HMMT-Feb

alg

Feb

Let $A$ denote the set of all integers $n$ such that $1 \leq n \leq 10000$, and moreover the sum of the decimal digits of $n$ is 2. Find the sum of the squares of the elements of $A$.

7294927

From the given conditions, we want to calculate \[ \sum_{i=0}^{3} \sum_{j=i}^{3}\left(10^{i}+10^{j}\right)^{2} \] By observing the formula, we notice that each term is an exponent of 10 . $10^{6}$ shows up 7 times, $10^{5}$ shows up 2 times, $10^{4}$ shows up 9 times, $10^{3}$ shows up 4 times, $10^{2}$ shows up 9 times, 10 shows 2 times, 1 shows up 7 times. Thus the answer is 7294927 . $\fbox{7294927}$.

HMMT Feb Easy

HMMT-Feb Algebra

HMMT

HMMT-Nov

guts

Nov

Let $A B C D$ be a quadrilateral with $A=(3,4), B=(9,-40), C=(-5,-12), D=(-7,24)$. Let $P$ be a point in the plane (not necessarily inside the quadrilateral). Find the minimum possible value of $A P+B P+C P+D P$.

16 \sqrt{17}+8 \sqrt{5}

By the triangle inequality, $A P+C P \geq A C$ and $B P+D P \geq B D$. So $P$ should be on $A C$ and $B D$; i.e. it should be the intersection of the two diagonals. Then $A P+B P+C P+D P=A C+B D$, which is easily computed to be $16 \sqrt{17}+8 \sqrt{5}$ by the Pythagorean theorem. Note that we require the intersection of the diagonals to actually exist for this proof to work, but $A B C D$ is convex and this is not an issue. $\fbox{16 \sqrt{17}+8 \sqrt{5}}$.

HMMT Nov Guts

HMMT-Nov Guts

AMC

AMC10

10A

N/A

Two circles of radius $5$ are externally tangent to each other and are internally tangent to a circle of radius $13$ at points $A$ and $B$, as shown in the diagram. The distance $AB$ can be written in the form $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$? [asy] draw(circle((0,0),13)); draw(circle((5,-6.2),5)); draw(circle((-5,-6.2),5)); label("$B$", (9.5,-9.5), S); label("$A$", (-9.5,-9.5), S); [/asy]

69

[asy] draw(circle((0,0),13)); draw(circle((5,-6.25),5)); draw(circle((-5,-6.25),5)); label("$A$", (-8.125,-10.15), S); label("$B$", (8.125,-10.15), S); draw((0,0)--(-8.125,-10.15)); draw((0,0)--(8.125,-10.15)); draw((-5,-6.25)--(5,-6.25)); draw((-8.125,-10.15)--(8.125,-10.15)); label("$X$", (0,0), N); label("$Y$", (-5,-6.25),NW); label("$Z$", (5,-6.25),NE); [/asy] Let the center of the surrounding circle be $X$. The circle that is tangent at point $A$ will have point $Y$ as the center. Similarly, the circle that is tangent at point $B$ will have point $Z$ as the center. Connect $AB$, $YZ$, $XA$, and $XB$. Now observe that $\triangle XYZ$ is similar to $\triangle XAB$ by SAS. Writing out the ratios, we get \[\frac{XY}{XA}=\frac{YZ}{AB} \Rightarrow \frac{13-5}{13}=\frac{5+5}{AB} \Rightarrow \frac{8}{13}=\frac{10}{AB} B=\frac{65}{4}.\] Therefore, our answer is $65+4= \fbox{69}$.

AMC10 Second Half

AMC10 A

HMMT

HMMT-Nov

guts

Nov

Kelvin the frog lives in a pond with an infinite number of lily pads, numbered $0,1,2,3$, and so forth. Kelvin starts on lily pad 0 and jumps from pad to pad in the following manner: when on lily pad $i$, he will jump to lily pad $(i+k)$ with probability $\frac{1}{2^{k}}$ for $k>0$. What is the probability that Kelvin lands on lily pad 2019 at some point in his journey?

\frac{1}{2}

Suppose we combine all of the lily pads with numbers greater than 2019 into one lily pad labeled $\infty$. Also, let Kelvin stop once he reaches one of these lily pads. Now at every leap, Kelvin has an equal chance of landing on 2019 as landing on $\infty$. Furthermore, Kelvin is guaranteed to reach 2019 or $\infty$ within 2020 leaps. Therefore the chance of landing on 2019 is the same as missing it, so our answer is just $\frac{1}{2}$. $\fbox{\frac{1}{2}}$.

HMMT Nov Guts

HMMT-Nov Guts

HMMT

HMMT-Feb

alg

Feb

Let $N=2^{\left(2^{2}\right)}$ and $x$ be a real number such that $N^{\left(N^{N}\right)}=2^{\left(2^{x}\right)}$. Find $x$.

66

We compute \[ N^{\left(N^{N}\right)}=16^{16^{16}}=2^{4 \cdot 2^{4 \cdot 2^{4}}}=2^{2^{2^{6}+2}}=2^{2^{66}} \] so $x=66$. $\fbox{66}$.

HMMT Feb Easy

HMMT-Feb Algebra

AIME

AIME

II

N/A

Abe can paint the room in 15 hours, Bea can paint 50 percent faster than Abe, and Coe can paint twice as fast as Abe. Abe begins to paint the room and works alone for the first hour and a half. Then Bea joins Abe, and they work together until half the room is painted. Then Coe joins Abe and Bea, and they work together until the entire room is painted. Find the number of minutes after Abe begins for the three of them to finish painting the room.

334

From the given information, we can see that Abe can paint $\frac{1}{15}$ of the room in an hour, Bea can paint $\frac{1}{15}\times\frac{3}{2} = \frac{1}{10}$ of the room in an hour, and Coe can paint the room in $\frac{1}{15}\times 2 = \frac{2}{15}$ of the room in an hour. After $90$ minutes, Abe has painted $\frac{1}{15}\times\frac{3}{2}=\frac{1}{10}$ of the room. Working together, Abe and Bea can paint $\frac{1}{15}+\frac{1}{10}=\frac{1}{6}$ of the room in an hour, so it takes then $\frac{2}{5}\div \frac{1}{6}= \frac{12}{5}$ hours to finish the first half of the room. All three working together can paint $\frac{1}{6}+\frac{2}{15}=\frac{3}{10}$ of the room in an hour, and it takes them $\frac{1}{2}\div \frac{3}{10}=\frac{5}{3}$ hours to finish the room. The total amount of time they take is \[90+\frac{12}{5}\times 60+\frac{5}{3}\times 60 = 90+ 144 + 100 = \fbox{334} \text{\ minutes.}\]

Easy AIME Problems

AIME

HMMT

HMMT-Feb

guts

Feb

Let $f(x)=x^{2}+a x+b$ and $g(x)=x^{2}+c x+d$ be two distinct real polynomials such that the $x$-coordinate of the vertex of $f$ is a root of $g$, the $x$-coordinate of the vertex of $g$ is a root of $f$ and both $f$ and $g$ have the same minimum value. If the graphs of the two polynomials intersect at the point $(2012,-2012)$, what is the value of $a+c$ ?

-8048

It is clear, by symmetry, that 2012 is the equidistant from the vertices of the two quadratics. Then it is clear that reflecting $f$ about the line $x=2012$ yields $g$ and vice versa. Thus the average of each pair of roots is 2012. Thus the sum of the four roots of $f$ and $g$ is 8048 , so $a+c=-8048$. $\fbox{-8048}$.

HMMT Feb Guts

HMMT-Feb Guts

HMMT

HMMT-Nov

guts

Nov

Let $r_{1}, r_{2}, r_{3}, r_{4}$ be the four roots of the polynomial $x^{4}-4 x^{3}+8 x^{2}-7 x+3$. Find the value of \[ \frac{r_{1}^{2}}{r_{2}^{2}+r_{3}^{2}+r_{4}^{2}}+\frac{r_{2}^{2}}{r_{1}^{2}+r_{3}^{2}+r_{4}^{2}}+\frac{r_{3}^{2}}{r_{1}^{2}+r_{2}^{2}+r_{4}^{2}}+\frac{r_{4}^{2}}{r_{1}^{2}+r_{2}^{2}+r_{3}^{2}} \]

-4

Add 1 to each fraction to get \[ \frac{r_{1}^{2}+r_{2}^{2}+r_{3}^{2}+r_{4}^{2}}{r_{2}^{2}+r_{3}^{2}+r_{4}^{2}}+\frac{r_{1}^{2}+r_{2}^{2}+r_{3}^{2}+r_{4}^{2}}{r_{1}^{2}+r_{3}^{2}+r_{4}^{2}}+\frac{r_{1}^{2}+r_{2}^{2}+r_{3}^{2}+r_{4}^{2}}{r_{1}^{2}+r_{2}^{2}+r_{4}^{2}}+\frac{r_{1}^{2}+r_{2}^{2}+r_{3}^{2}+r_{4}^{2}}{r_{1}^{2}+r_{2}^{2}+r_{3}^{2}} \] This seems like a difficult problem until one realizes that \[ r_{1}^{2}+r_{2}^{2}+r_{3}^{2}+r_{4}^{2}=\left(r_{1}+r_{2}+r_{3}+r_{4}\right)^{2}-2\left(r_{1} r_{2}+r_{1} r_{3}+r_{1} r_{4}+r_{2} r_{3}+r_{2} r_{4}+r_{3} r_{4}\right)=4^{2}-2 \cdot 8=0 \] Thus, our current expression is 0 . Noting that we added 4, the original value had to be -4 . $\fbox{-4}$.

HMMT Nov Guts

HMMT-Nov Guts

HMMT

HMMT-Nov

guts

Nov

A positive integer $\overline{A B C}$, where $A, B, C$ are digits, satisfies \[ \overline{A B C}=B^{C}-A \] Find $\overline{A B C}$.

127

The equation is equivalent to $100 A+10 B+C=B^{C}-A$. Suppose $A=0$, so that we get $10 B+C=B^{C}$. Reducing $\bmod B$, we find that $C$ must be divisible by $B$. $C \neq 0$, since otherwise $10 B=1$, contradiction, so $C \geq B$. Thus $10 B+C \geq B^{B}$ for digits $B, C$. For $B \geq 4$, we have $100>10 B+C \geq B^{B}>100$, a contradiction, so $B=1,2,3$. We can easily test that these do not yield solutions, so there are no solutions when $A=0$. Thus $A \geq 1$, and so $100 \leq 100 A+10 B+C \leq 1000$, and thus $100 \leq B^{C}-A \leq 1000.1 \leq A \leq 10$, so we have $101 \leq B^{C} \leq 1010$. We can test that the only pairs $(B, C)$ that satisfy this condition are $(2,7),(2,8),(2,9),(3,5),(3,6),(4,4),(5,3),(6,3),(7,3),(8,3),(9,3)$. Of these pairs, only $(2,7)$ yields a solution to the original equation, namely $A=1, B=2, C=7$. Thus $\overline{A B C}=127$. $\fbox{127}$.

HMMT Nov Guts

HMMT-Nov Guts

HMMT

HMMT-Feb

comb

Feb

Ava and Tiffany participate in a knockout tournament consisting of a total of 32 players. In each of 5 rounds, the remaining players are paired uniformly at random. In each pair, both players are equally likely to win, and the loser is knocked out of the tournament. The probability that Ava and Tiffany play each other during the tournament is $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers. Compute $100 a+b$.

116

Solution: Each match eliminates exactly one player, so exactly $32-1=31$ matches are played, each of which consists of a different pair of players. Among the $\left(\begin{array}{c}32 \\ 2\end{array}\right)=\frac{32 \cdot 31}{2}=496$ pairs of players, each pair is equally likely to play each other at some point during the tournament. Therefore, the probability that Ava and Tiffany form one of the 31 pairs of players that play each other is $\frac{31}{496}=\frac{1}{16}$, giving an answer of $100 \cdot 1+16=116$. $\fbox{116}$.

HMMT Feb Easy

HMMT-Feb Combinatorics

AMC

AMC10

10B

N/A

In $\triangle ABC$, $AB=6$, $AC=8$, $BC=10$, and $D$ is the midpoint of $\overline{BC}$. What is the sum of the radii of the circles inscribed in $\triangle ADB$ and $\triangle ADC$?

\frac{17}{6}

We note that by the converse of the Pythagorean Theorem, $\triangle ABC$ is a right triangle with a right angle at $A$. Therefore, $AD = BD = CD = 5$, and $[ADB] = [ADC] = 12$. Since $A = rs,$ we have $r = \frac As$, so the inradius of $\triangle ADB$ is $\frac{12}{(5+5+6)/2} = \frac 32$, and the inradius of $\triangle ADC$ is $\frac{12}{(5+5+8)/2} = \frac 43$. Adding the two together, we have $\fbox{\frac{17}{6}}$.

AMC10 Final Problems

AMC10 B

AIME

AIME

I

N/A

Given $\triangle ABC$ and a point $P$ on one of its sides, call line $\ell$ the $\textit{splitting line}$ of $\triangle ABC$ through $P$ if $\ell$ passes through $P$ and divides $\triangle ABC$ into two polygons of equal perimeter. Let $\triangle ABC$ be a triangle where $BC = 219$ and $AB$ and $AC$ are positive integers. Let $M$ and $N$ be the midpoints of $\overline{AB}$ and $\overline{AC},$ respectively, and suppose that the splitting lines of $\triangle ABC$ through $M$ and $N$ intersect at $30^\circ.$ Find the perimeter of $\triangle ABC.$

459

Denote $BC = a$, $CA = b$, $AB = c$. Let the splitting line of $\triangle ABC$ through $M$ (resp. $N$) crosses $\triangle ABC$ at another point $X$ (resp. $Y$). WLOG, we assume $c \leq b$. $\textbf{Case 1}$: $a \leq c \leq b$. We extend segment $AB$ to $D$, such that $BD = a$. We extend segment $AC$ to $E$, such that $CE = a$. In this case, $X$ is the midpoint of $AE$, and $Y$ is the midpoint of $AD$. Because $M$ and $X$ are the midpoints of $AB$ and $AE$, respectively, $MX \parallel BE$. Because $N$ and $Y$ are the midpoints of $AC$ and $AD$, respectively, $NY \parallel CD$. Because $CB = CE$, $\angle CBE =\angle CEB = \frac{\angle ACB}{2}$. Because $BC = BD$, $\angle BCD = \angle BDC = \frac{\angle ABC}{2}$. Let $BE$ and $CD$ intersect at $O$. Because $MX \parallel BE$ and $NY \parallel CD$, the angle formed between lines $MX$ and $NY$ is congruent to $\angle BOD$. Hence, $\angle BOD = 30^\circ$ or $150^\circ$. We have \begin{align} \angle BOD & = \angle CBE + \angle BCD \\ & = \frac{\angle ACB}{2} + \frac{\angle ABC}{2} \\ & = 90^\circ - \frac{\angle A}{2} . \end{align} Hence, we must have $\angle BOD = 30^\circ$, not $150^\circ$. Hence, $\angle A = 120^\circ$. This implies $a > b$ and $a >c$. This contradicts the condition specified for this case. Therefore, this case is infeasible. $\textbf{Case 2}$: $c \leq a \leq b$. We extend segment $CB$ to $D$, such that $BD = c$. We extend segment $AC$ to $E$, such that $CE = a$. In this case, $X$ is the midpoint of $AE$, and $Y$ is the midpoint of $CD$. Because $M$ and $X$ are the midpoints of $AB$ and $AE$, respectively, $MX \parallel BE$. Because $N$ and $Y$ are the midpoints of $AC$ and $CD$, respectively, $NY \parallel AD$. Because $CB = CE$, $\angle CBE =\angle CEB = \frac{\angle ACB}{2}$. Because $BA = BD$, $\angle BAD = \angle BDA = \frac{\angle ABC}{2}$. Let $O$ be a point of $AC$, such that $BO \parallel AD$. Hence, $\angle OBC = \angle BDA = \frac{B}{2}$. Because $MX \parallel BE$ and $NY \parallel AD$ and $AD \parallel BO$, the angle formed between lines $MX$ and $NY$ is congruent to $\angle OBE$. Hence, $\angle OBE = 30^\circ$ or $150^\circ$. We have \begin{align} \angle OBE & = \angle OBC + \angle CBE \\ & = \frac{\angle ABC}{2} + \frac{\angle ACB}{2} \\ & = 90^\circ - \frac{\angle A}{2} . \end{align} Hence, we must have $\angle OBE = 30^\circ$, not $150^\circ$. Hence, $\angle A = 120^\circ$. This implies $a > b$ and $a >c$. This contradicts the condition specified for this case. Therefore, this case is infeasible. $\textbf{Case 3}$: $c \leq b \leq a$. We extend segment $CB$ to $D$, such that $BD = c$. We extend segment $BC$ to $E$, such that $CE = b$. In this case, $X$ is the midpoint of $BE$, and $Y$ is the midpoint of $CD$. Because $M$ and $X$ are the midpoints of $AB$ and $BE$, respectively, $MX \parallel AE$. Because $N$ and $Y$ are the midpoints of $AC$ and $CD$, respectively, $NY \parallel AD$. Because $CA = CE$, $\angle CAE =\angle CEB = \frac{\angle ACB}{2}$. Because $BA = BD$, $\angle BAD = \angle BDA = \frac{\angle ABC}{2}$. Because $MX \parallel AE$ and $NY \parallel AD$, the angle formed between lines $MX$ and $NY$ is congruent to $\angle DAE$. Hence, $\angle DAE = 30^\circ$ or $150^\circ$. We have \begin{align} \angle DAE & = \angle BAD + \angle CAE + \angle BAC \\ & = \frac{\angle ABC}{2} + \frac{\angle ACB}{2} + \angle BAC \\ & = 90^\circ + \frac{\angle BAC}{2} . \end{align} Hence, we must have $\angle OBE = 150^\circ$, not $30^\circ$. Hence, $\angle BAC = 120^\circ$. In $\triangle ABC$, by applying the law of cosines, we have \begin{align} a^2 & = b^2 + c^2 - 2bc \cos \angle BAC\\ & = b^2 + c^2 - 2bc \cos 120^\circ \\ & = b^2 + c^2 + bc . \end{align} Because $a = 219$, we have \[ b^2 + c^2 + bc = 219^2 . \] Now, we find integer solution(s) of this equation with $c \leq b$. Multiplying this equation by 4, we get \[ \left( 2 c + b \right)^2 + 3 b^2 = 438^2 . \hspace{1cm} (1) \] Denote $d = 2 c + b$. Because $c \leq b$, $b < d \leq 3 b$. Because $438^2 - 3 b^2 \equiv 0 \pmod{3}$, $d^2 \equiv 0 \pmod{3}$. Thus, $d \equiv 0 \pmod{3}$. This implies $d^2 \equiv 0 \pmod{9}$. We also have $438^2 \equiv 0 \pmod{9}$. Hence, $3 b^2 \equiv 0 \pmod{9}$. This implies $b \equiv 0 \pmod{3}$. Denote $b = 3 p$ and $d = 3 q$. Hence, $p < q \leq 3 p$. Hence, Equation (1) can be written as \[ q^2 + 3 p^2 = 146^2 . \hspace{1cm} (2) \] Now, we solve this equation. First, we find an upper bound of $q$. We have $q^2 + 3 p^2 \geq q^2 + 3 \left( \frac{q}{3} \right)^2 = \frac{4 q^2}{3}$. Hence, $\frac{4 q^2}{3} \leq 146^2$. Hence, $q \leq 73 \sqrt{3} < 73 \cdot 1.8 = 131.4$. Because $q$ is an integer, we must have $q \leq 131$. Second, we find a lower bound of $q$. We have $q^2 + 3 p^2 < q^2 + 3 q^2 = 4 q^2$. Hence, $4 q^2 > 146^2$. Hence, $q > 73$. Because $q$ is an integer, we must have $q \geq 74$. Now, we find the integer solutions of $p$ and $q$ that satisfy Equation (2) with $74 \leq q \leq 131$. First, modulo 9, \begin{align} q^2 & \equiv 146^2 - 3 p^2 \\ & \equiv 4 - 3 \cdot ( 0 \mbox{ or } 1 ) \\ & \equiv 4 \mbox{ or } 1 . \end{align} Hence $q \equiv \pm 1, \pm 2 \pmod{9}$. Second, modulo 5, \begin{align} q^2 & \equiv 146^2 - 3 p^2 \\ & \equiv 1 + 2 p^2 \\ & \equiv 1 + 2 \cdot ( 0 \mbox{ or } 1 \mbox{ or } -1 ) \\ & \equiv 1 \mbox{ or } 3 \mbox{ or } - 1 . \end{align} Because $q^2 \equiv 0 \mbox{ or } 1 \mbox{ or } - 1$, we must have $q^2 \equiv 1 \mbox{ or } - 1$. Hence, $5 \nmid q$. Third, modulo 7, \begin{align} q^2 & \equiv 146^2 - 3 p^2 \\ & \equiv 1 - 3 \cdot ( 0 \mbox{ or } 1 \mbox{ or } 5 \mbox{ or } 2 ) \\ & \equiv 1 \mbox{ or } 2 \mbox{ or } 3 \mbox{ or } 5 . \end{align} Because $q^2 \equiv 0 \mbox{ or } 1 \mbox{ or } 2 \mbox{ or } 4 \pmod{ 7 }$, we must have $q^2 \equiv 1 \mbox{ or } 2 \pmod{7}$. Hence, $q \equiv 1, 3, 4, 6 \pmod{7}$. Given all conditions above, the possible $q$ are 74, 83, 88, 92, 97, 101, 106, 109, 116, 118, 127. By testing all these numbers, we find that the only solution is $q = 97$. This implies $p = 63$. Hence, $b = 3p = 189$ and $d = 3q = 291$. Hence, $c = \frac{d - b}{2} = 51$. Therefore, the perimeter of $\triangle ABC$ is $b + c + a = 189 + 51 + 219 = \fbox{459}$. ~Steven Chen (www.professorchenedu.com)

Very Hard AIME Problems

AIME

HMMT

HMMT-Feb

geo

Feb

Jarris the triangle is playing in the $(x, y)$ plane. Let his maximum $y$ coordinate be $k$. Given that he has side lengths 6,8 , and 10 and that no part of him is below the $x$-axis, find the minimum possible value of $k$.

\frac{24}{5}

By playing around, we find that Jarris should have his hypotenuse flat on the $x$ axis. The desired minimum value of $k$ is then the length of the altitude to the hypotenuse. Thus, by computing the area of the triangle in two ways, $\frac{1}{2} \cdot 10 \cdot k=\frac{1}{2} \cdot 6 \cdot 8$ and so $k=\frac{24}{5}$. $\fbox{\frac{24}{5}}$.

HMMT Feb Easy

HMMT-Feb Geometry

HMMT

HMMT-Nov

guts

Nov

Define a sequence $\left\{a_{n}\right\}$ by $a_{1}=1$ and $a_{n}=\left(a_{n-1}\right) !+1$ for every $n>1$. Find the least $n$ for which $a_{n}>10^{10}$.

6

We have $a_{2}=2, a_{3}=3, a_{4}=7, a_{5}=7 !+1=5041$, and $a_{6}=5041 !+1$. But \[ 5041 !+1 \gg 5041 \cdot 5040 \cdot 5039>10^{10} \] Hence, the answer is 6 . $\fbox{6}$.

HMMT Nov Guts

HMMT-Nov Guts

AMC

AMC8

8

N/A

Rectangle $ABCD$ is inscribed in a semicircle with diameter $\overline{FE},$ as shown in the figure. Let $DA=16,$ and let $FD=AE=9.$ What is the area of $ABCD?$ [asy] draw(arc((0,0),17,180,0)); draw((-17,0)--(17,0)); fill((-8,0)--(-8,15)--(8,15)--(8,0)--cycle, 1.5*grey); draw((-8,0)--(-8,15)--(8,15)--(8,0)--cycle); dot("$A$",(8,0), 1.25*S); dot("$B$",(8,15), 1.25*N); dot("$C$",(-8,15), 1.25*N); dot("$D$",(-8,0), 1.25*S); dot("$E$",(17,0), 1.25*S); dot("$F$",(-17,0), 1.25*S); label("$16$",(0,0),N); label("$9$",(12.5,0),N); label("$9$",(-12.5,0),N); [/asy]

240

[asy] draw(arc((0,0),17,180,0)); draw((-17,0)--(17,0)); fill((-8,0)--(-8,15)--(8,15)--(8,0)--cycle, 1.5*grey); draw((-8,0)--(-8,15)--(8,15)--(8,0)--cycle); dot("$A$",(8,0), 1.25*S); dot("$B$",(8,15), 1.25*N); dot("$C$",(-8,15), 1.25*N); dot("$D$",(-8,0), 1.25*S); dot("$E$",(17,0), 1.25*S); dot("$F$",(-17,0), 1.25*S); label("$16$",(0,0),N); label("$9$",(12.5,0),N); label("$9$",(-12.5,0),N); dot("$O$", (0,0), 1.25*S); draw((0,0)--(-8,15));[/asy] Let $O$ be the center of the semicircle. The diameter of the semicircle is $9+16+9=34$, so $OC = 17$. By symmetry, $O$ is the midpoint of $DA$, so $OD=OA=\frac{16}{2}= 8$. By the Pythagorean theorem in right-angled triangle $ODC$ (or $OBA$), we have that $CD$ (or $AB$) is $\sqrt{17^2-8^2}=15$. Accordingly, the area of $ABCD$ is $16\cdot 15=\fbox{240}$.

AMC8 Second Half

AMC8

AMC

AMC12

12B

N/A

Let $ABCD$ be a parallelogram with area $15$. Points $P$ and $Q$ are the projections of $A$ and $C,$ respectively, onto the line $BD;$ and points $R$ and $S$ are the projections of $B$ and $D,$ respectively, onto the line $AC.$ See the figure, which also shows the relative locations of these points. [asy] size(350); defaultpen(linewidth(0.8)+fontsize(11)); real theta = aTan(1.25/2); pair A = 2.5*dir(180+theta), B = (3.35,0), C = -A, D = -B, P = foot(A,B,D), Q = -P, R = foot(B,A,C), S = -R; draw(A--B--C--D--A^^B--D^^R--S^^rightanglemark(A,P,D,6)^^rightanglemark(C,Q,D,6)); draw(B--R^^C--Q^^A--P^^D--S,linetype("4 4")); dot("$A$",A,dir(270)); dot("$B$",B,E); dot("$C$",C,N); dot("$D$",D,W); dot("$P$",P,SE); dot("$Q$",Q,NE); dot("$R$",R,N); dot("$S$",S,dir(270)); [/asy] Suppose $PQ=6$ and $RS=8,$ and let $d$ denote the length of $\overline{BD},$ the longer diagonal of $ABCD.$ Then $d^2$ can be written in the form $m+n\sqrt p,$ where $m,n,$ and $p$ are positive integers and $p$ is not divisible by the square of any prime. What is $m+n+p?$

81

Let $X$ denote the intersection point of the diagonals $AC$ and $BD$. Remark that by symmetry $X$ is the midpoint of both $\overline{PQ}$ and $\overline{RS}$, so $XP = XQ = 3$ and $XR = XS = 4$. Now note that since $\angle APB = \angle ARB = 90^\circ$, quadrilateral $ARPB$ is cyclic, and so \[XR\cdot XA = XP\cdot XB,\]which implies $\tfrac{XA}{XB} = \tfrac{XP}{XR} = \tfrac34$. Thus let $x> 0$ be such that $XA = 3x$ and $XB = 4x$. Then Pythagorean Theorem on $\triangle APX$ yields $AP = \sqrt{AX^2 - XP^2} = 3\sqrt{x^2-1}$, and so\[[ABCD] = 2[ABD] = AP\cdot BD = 3\sqrt{x^2-1}\cdot 8x = 24x\sqrt{x^2-1}=15\]Solving this for $x^2$ yields $x^2 = \tfrac12 + \tfrac{\sqrt{41}}8$, and so\[(8x)^2 = 64x^2 = 64\left(\tfrac12 + \tfrac{\sqrt{41}}8\right) = 32 + 8\sqrt{41}.\]The requested answer is $32 + 8 + 41 = \fbox{81}$.

AMC12 Final Problems

AMC12 B

AMC

AMC10

10A

N/A

Jacob uses the following procedure to write down a sequence of numbers. First he chooses the first term to be 6. To generate each succeeding term, he flips a fair coin. If it comes up heads, he doubles the previous term and subtracts 1. If it comes up tails, he takes half of the previous term and subtracts 1. What is the probability that the fourth term in Jacob's sequence is an integer?

\frac{5}{8}

We can see that as long as the last flip is heads, it will be an integer, so there is a $\dfrac{1}{2}$ chance of this happening. Doing a little casework, we see that the only possibility when the last flip is tails is when the third flip brings it to 0. There is a $\dfrac{1}{8}$ chance of this happening. Therefore, there is a $\dfrac{1}{2}+\dfrac{1}{8}=\fbox{\frac{5}{8}}$ possibility of ending with an integer.

AMC10 Final Problems

AMC10 A

AMC

AMC10

10A

Nov

How many ordered pairs $(x,y)$ of real numbers satisfy the following system of equations? \begin{align} x^2+3y&=9 \\ (|x|+|y|-4)^2 &= 1 \end{align}

5

The second equation is $(|x|+|y| - 4)^2 = 1$. We know that the graph of $|x| + |y|$ is a very simple diamond shape, so let's see if we can reduce this equation to that form: \[(|x|+|y| - 4)^2 = 1 \implies |x|+|y| - 4 = \pm 1 \implies |x|+|y| = \{3,5\}.\] We now have two separate graphs for this equation and one graph for the first equation, so let's put it on the coordinate plane: [asy] Label f; f.p=fontsize(6); xaxis(-8,8,Ticks(f, 1.0,0.5)); yaxis(-8,8,Ticks(f, 1.0,0.5)); real f(real x) { return 3-x; } draw(graph(f,0,3)); real f(real x) { return 3+x; } draw(graph(f,0,-3)); real f(real x) { return x-3; } draw(graph(f,0,3)); real f(real x) { return -x-3; } draw(graph(f,0,-3)); real f(real x) { return 5-x; } draw(graph(f,0,5)); real f(real x) { return 5+x; } draw(graph(f,0,-5)); real f(real x) { return x-5; } draw(graph(f,0,5)); real f(real x) { return -x-5; } draw(graph(f,0,-5)); real f(real x) { return 3-x; } draw(graph(f,0,3)); real f(real x) { return 3+x; } draw(graph(f,0,-3)); real f(real x) { return x-3; } draw(graph(f,0,3)); real f(real x) { return (-x^2)/3+3; } draw(graph(f,-5,5)); [/asy] We see from the graph that there are $5$ intersections, so the answer is $\fbox{5}$.

AMC10 Second Half

AMC10 A

HMMT

HMMT-Feb

guts

Feb

From the point $(x, y)$, a legal move is a move to $\left(\frac{x}{3}+u, \frac{y}{3}+v\right)$, where $u$ and $v$ are real numbers such that $u^{2}+v^{2} \leq 1$. What is the area of the set of points that can be reached from $(0,0)$ in a finite number of legal moves?

\frac{9 \pi}{4}

We claim that the set of points is the disc with radius $\frac{3}{2}$ centered at the origin, which clearly has area $\frac{9 \pi}{4}$. First, we show that the set is contained in this disc. This is because if we are currently at a distance of $r$ from the origin, then we can't end up at a distance of greater than $\frac{r}{3}+1$ from the origin after a single move. Since $\frac{r}{3}+1<\frac{3}{2}$ if $r<\frac{3}{2}$, we will always end up in the disc of radius $\frac{3}{2}$ if we start in it. Since the origin is inside this disc, any finite number of moves will leave us inside this disc. Next, we show that all points in this disc can be reached in a finite number of moves. Indeed, after one move we can get all points within a distance of 1 . After two moves, we can get all points within a distance of $\frac{4}{3}$. After three moves, we can get all points within a distance of $\frac{13}{9}$. In general, after $n$ moves we can get all points within a distance of $\frac{3}{2}-\frac{1}{2 \cdot 3^{k-1}}$. This means that for any distance $d<\frac{3}{2}$, we will eventually get all points within a distance of $d$, so all points in the disc of radius $\frac{3}{2}$ can be reached after some number of moves. $\fbox{\frac{9 \pi}{4}}$.

HMMT Feb Guts

HMMT-Feb Guts

AMC

AMC12

12B

N/A

For real numbers $a$ and $b$, define $a\textdollar b = (a - b)^2$. What is $(x - y)^2\textdollar(y - x)^2$?

0

$\left[ (x-y)^2 - (y-x)^2 \right]^2$ $\left[ (x-y)^2 - (x-y)^2 \right]^2$ $[0]^2$ $0 \Rightarrow $ $\fbox{0}$.

AMC12 First Half

AMC12 B