ChenDRAG/VeRL_math_validation huggingface dataset:
Problem: Let an be a sequence such that $a_0 = 0$ and:
$a_{3n+1} = a_{3n} + 1 = a_n + 1$
$a_{3n+2} = a_{3n} + 2 = a_n + 2$
for all natural numbers $n$. How many $n$ less than $2012$ have the property that $a_n = 7$?
Answer: 379
Problem: Jim has two fair $6$-sided dice, one whose faces are labelled from $1$ to $6$, and the second whose faces are labelled from $3$ to $8$. Twice, he randomly picks one of the dice (each die equally likely) and rolls it. Given the sum of the resulting two rolls is $9$, if $\frac{m}{n}$ is the probability he rolled the same die twice where $m, n$ are relatively prime positive integers, then what is $m + n$?
Answer: 7
Problem: A square is inscribed in an ellipse such that two sides of the square respectively pass through the two foci of the ellipse. The square has a side length of $4$. The square of the length of the minor axis of the ellipse can be written in the form $a + b\sqrt{c}$ where $a, b$, and $c$ are integers, and $c$ is not divisible by the square of any prime. Find the sum $a + b + c$.
Answer: 21
Problem: Proctors Andy and Kristin have a PUMaC team of eight students labelled $s_1, s_2, ... , s_8$ (the PUMaC staff being awful with names). The following occurs:
$1$. Andy tells the students to arrange themselves in a line in arbitrary order.
$2$. Kristin tells each student $s_i$ to move to the current spot of student $s_j$ , where $j \equiv 3i + 1$ (mod $8$).
$3$. Andy tells each student $s_i$ to move to the current spot of the student who was in the $i$th position of the line after step $1$.
How many possible orders can the students be in now?
Answer: 1260
Problem: During chemistry labs, we oftentimes fold a disk-shaped filter paper twice, and then open up a flap of the quartercircle to form a cone shape, as in the diagram. What is the angle $\theta$, in degrees, of the bottom of the cone when we look at it from the side?
[img]https://cdn.artofproblemsolving.com/attachments/d/2/f8e3a7afb606dfd6fad277f547b116566a4a91.png[/img]
Answer: 60
Problem: If $x, y$, and $z$ are real numbers with $\frac{x - y}{z}+\frac{y - z}{x}+\frac{z - x}{y}= 36$, find $2012 +\frac{x - y}{z}\cdot \frac{y - z}{x}\cdot\frac{z - x}{y}$ .
Answer: 1976
Problem: A cyclic quadrilateral $ABCD$ has side lengths $AB = 3, BC = AD = 5$, and $CD = 8$. The radius of its circumcircle can be written in the form $a\sqrt{b}/c$, where $a, b, c$ are positive integers, $a, c$ are relatively prime, and $b$ is not divisible by the square of any prime. Find $a + b + c$.
Answer: 16
Problem: How many ways can $2^{2012}$ be expressed as the sum of four (not necessarily distinct) positive squares?
Answer: 1
Problem: A regular pentagon can have the line segments forming its boundary extended to lines, giving an arrangement of lines that intersect at ten points. How many ways are there to choose five points of these ten so that no three of the points are collinear?
Answer: 12
Problem: Find the number of pairs $(n,C)$ of positive integers such that $C\leq 100$ and $n^2+n+C$ is a perfect square.
Answer: 108
Problem: Suppose $P(x)$ is a degree $n$ monic polynomial with integer coefficients such that $2013$ divides $P(r)$ for exactly $1000$ values of $r$ between $1$ and $2013$ inclusive.  Find the minimum value of $n$.
Answer: 50
Problem: Let $A$ be the greatest possible value of a product of positive integers that sums to $2014$.  Compute the sum of all bases and exponents in the prime factorization of $A$.  For example, if $A=7\cdot 11^5$, the answer would be $7+11+5=23$.
Answer: 677
Problem: If two distinct integers from $1$ to $50$ inclusive are chosen at random, what is the expected value of their product?  Note: The expectation is defined as the sum of the products of probability and value, i.e., the expected value of a coin flip that gives you $\$10$ if head and $\$5$ if tail is $\tfrac12\times\$10+\tfrac12\times\$5=\$7.5$.
Answer: 646
Problem: How many positive integers $n$ less than $1000$ have the property that the number of positive integers less than $n$ which are coprime to $n$ is exactly $\tfrac n3$?
Answer: 24
Problem: A sequence of vertices $v_1,v_2,\ldots,v_k$ in a graph, where $v_i=v_j$ only if $i=j$ and $k$ can be any positive integer, is called a $\textit{cycle}$ if $v_1$ is attached by an edge to $v_2$, $v_2$ to $v_3$, and so on to $v_k$ connected to $v_1$.  Rotations and reflections are distinct: $A,B,C$ is distinct from $A,C,B$ and $B,C,A$.  Supposed a simple graph $G$ has $2013$ vertices and $3013$ edges.  What is the minimal number of cycles possible in $G$?
Answer: 6006
Problem: You roll three fair six-sided dice.  Given that the highest number you rolled is a $5$, the expected value of the sum of the three dice can be written as $\tfrac ab$ in simplest form.  Find $a+b$.
Answer: 706
Problem: Compute the smallest integer $n\geq 4$ such that $\textstyle\binom n4$ ends in $4$ or more zeroes (i.e. the rightmost four digits of $\textstyle\binom n4$ are $0000$).
Answer: 8128
Problem: Suppose $a,b,c>0$ are integers such that \[abc-bc-ac-ab+a+b+c=2013.\]  Find the number of possibilities for the ordered triple $(a,b,c)$.
Answer: 18
Problem: Betty Lou and Peggy Sue take turns flipping switches on a $100 \times 100$ grid. Initially, all switches are "off". Betty Lou always flips a horizontal row of switches on her turn; Peggy Sue always flips a vertical column of switches. When they finish, there is an odd number of switches turned "on'' in each row and column. Find the maximum number of switches that can be on, in total, when they finish.
Answer: 9802
Problem: (Following question 1) Now instead consider an infinite strip of squares, labeled with the integers $0, 1, 2, \ldots$ in that order. You start at the square labeled $0$. You want to end up at the square labeled $3$. In how many ways can this be done in exactly $15$ moves?
Answer: 2002
Problem: Let $x_1=\sqrt{10}$ and $y_1=\sqrt3$.  For all $n\geq 2$, let \begin{align*}x_n&=x_{n-1}\sqrt{77}+15y_{n-1}\\y_n&=5x_{n-1}+y_{n-1}\sqrt{77}\end{align*} Find $x_5^6+2x_5^4-9x_5^4y_5^2-12x_5^2y_5^2+27x_5^2y_5^4+18y_5^4-27y_5^6.$
Answer: 4608
Problem: A token is placed in the leftmost square in a strip of four squares. In each move, you are allowed to move the token left or right along the strip by sliding it a single square, provided that the token stays on the strip. In how many ways can the token be moved so that after exactly $15$ moves, it is in the rightmost square of the strip?
Answer: 377
Problem: Let $k$ be a positive integer with the following property: For every subset $A$ of $\{1,2,\ldots, 25\}$ with $|A|=k$, we can find distinct elements $x$ and $y$ of $A$ such that $\tfrac23\leq\tfrac xy\leq\tfrac 32$.  Find the smallest possible value of $k$.
Answer: 7
Problem: Find the number of pairs $(n,C)$ of positive integers such that $C\leq 100$ and $n^2+n+C$ is a perfect square.
Answer: 180
Problem: Including the original, how many ways are there to rearrange the letters in PRINCETON so that no two vowels (I, E, O) are consecutive and no three consonants (P, R, N, C, T, N) are consecutive?
Answer: 17280
Problem: Let $d$ be the greatest common divisor of $2^{30^{10}}-2$ and $2^{30^{45}}-2$.  Find the remainder when $d$ is divided by $2013$.
Answer: 2012
Problem: Draw an equilateral triangle with center $O$.  Rotate the equilateral triangle $30^\circ, 60^\circ, 90^\circ$ with respect to $O$ so there would be four congruent equilateral triangles on each other.  Look at the diagram.  If the smallest triangle has area $1$, the area of the original equilateral triangle could be expressed as $p+q\sqrt r$ where $p,q,r$ are positive integers and $r$ is not divisible by a square greater than $1$.  Find $p+q+r$.
Answer: 102
Problem: What is the smallest positive integer $n$ such that $2013^n$ ends in $001$ (i.e. the rightmost three digits of $2013^n$ are $001$?
Answer: 100
Problem: Find the sum of all positive integers $m$ such that $2^m$ can be expressed as a sum of four factorials (of positive integers).

Note: The factorials do not have to be distinct.  For example, $2^4=16$ counts, because it equals $3!+3!+2!+2!$.
Answer: 21
Problem: Triangle $ABC$ satisfies $\angle ABC=\angle ACB=78^\circ$.  Points $D$ and $E$ lie on $AB,AC$ and satisfy $\angle BCD=24^\circ$ and $\angle CBE=51^\circ$.  If $\angle BED=x^\circ$, find $x$.
Answer: 12
Problem: Find the smallest positive integer $x$ such that

[list]
[*] $x$ is $1$ more than a multiple of $3$,
[*] $x$ is $3$ more than a multiple of $5$,
[*] $x$ is $5$ more than a multiple of $7$,
[*] $x$ is $9$ more than a multiple of $11$, and
[*] $x$ is $2$ more than a multiple of $13$.[/list]
Answer: 12703
Problem: What is the largest positive integer that cannot be expressed as a sum of non-negative integer multiple of $13$, $17$, and $23$?
Answer: 84
Problem: Let $f(x)=1-|x|$.  Let \begin{align*}f_n(x)&=(\overbrace{f\circ \cdots\circ f}^{n\text{ copies}})(x)\\g_n(x)&=|n-|x| |\end{align*} Determine the area of the region bounded by the $x$-axis and the graph of the function $\textstyle\sum_{n=1}^{10}f(x)+\textstyle\sum_{n=1}^{10}g(x).$
Answer: 440
Problem: Suppose the function $\psi$ satisfies $\psi(1)=\sqrt{2+\sqrt{2+\sqrt2}}$ and $\psi(3x)+3\psi(x)=\psi(x)^3$ for all real $x$.  Determine the greatest integer less than $\textstyle\prod_{n=1}^{100}\psi(3^n)$.
Answer: 5792
Problem: Let $x_1=1/20$, $x_2=1/13$, and \[x_{n+2}=\dfrac{2x_nx_{n+1}(x_n+x_{n+1})}{x_n^2+x_{n+1}^2}\] for all integers $n\geq 1$.  Evaluate $\textstyle\sum_{n=1}^\infty(1/(x_n+x_{n+1}))$.
Answer: 23
Problem: Circle $w$ with center $O$ meets circle $\Gamma$ at $X,Y,$ and $O$ is on $\Gamma$.  Point $Z\in\Gamma$ lies outside $w$ such that $XZ=11$, $OZ=15$, and $YZ=13$.  If the radius of circle $w$ is $r$, find $r^2$.
Answer: 82
Problem: The equation $x^5-2x^4-1=0$ has five complex roots $r_1,r_2,r_3,r_4,r_5$.  Find the value of \[\dfrac1{r_1^8}+\dfrac1{r_2^8}+\dfrac1{r_3^8}+\dfrac1{r_4^8}+\dfrac1{r_5^8}.\]
Answer: 16
Problem: Eight all different sushis are placed evenly on the edge of a round table, whose surface can rotate around the center. Eight people also evenly sit around the table, each with one sushi in front. Each person has one favorite sushi among these eight, and they are all distinct. They find that no matter how they rotate the table, there are never more than three people who have their favorite sushis in front of them simultaneously. By this requirement, how many different possible arrangements of the eight sushis are there? Two arrangements that differ by a rotation are considered the same.
Answer: 4274
Problem: Let $\mathcal{S}$ be the set of permutations of $\{1,2,\ldots,6\}$, and let $\mathcal{T}$ be the set of permutations of $\mathcal{S}$ that preserve compositions: i.e., if $F\in\mathcal{T}$ then \[F(f_2\circ f_1)=F(f_2)\circ F(f_1)\] for all $f_1,f_2\in\mathcal{S}$.  Find the number of elements $F\in\mathcal{T}$ such that if $f\in\mathcal{S}$ satisfies $f(1)=2$ and $f(2)=1$, then $(F(f))(1)=2$ and $(F(f))(2)=1$.
Answer: 192
Problem: Mereduth has many red boxes and many blue boxes. Coloon has placed five green boxes in a row on the ground, and Mereduth wants to arrange some number of her boxes on top of his row. Assume that each box must be placed so that it straddles two lower boxes. Including the one with no boxes, how many arrangements can Mereduth make?
Answer: 3113
Problem: Suppose $a,b$ are nonzero integers such that two roots of $x^3+ax^2+bx+9a$ coincide, and all three roots are integers.  Find $|ab|$.
Answer: 1344
Problem: Find the smallest positive integer $n$ with the following property:  for every sequence of positive integers $a_1,a_2,\ldots , a_n$ with $a_1+a_2+\ldots +a_n=2013$, there exist some (possibly one) consecutive term(s) in the sequence that add up to $70$.
Answer: 1033
Problem: You are eating at a fancy restaurant with a person you wish to impress. For some reason, you think that eating at least one spicy course and one meat-filled course will impress the person. The meal is five courses, with four options for each course. Each course has one option that is spicy and meat-filled, one option that is just spicy, one that is just meat-filled, and one that is neither spicy nor meat-filled. How many possible meals can you have?
Answer: 961
Problem: Consider all planes through the center of a $2\times2\times2$ cube that create cross sections that are regular polygons.  The sum of the cross sections for each of these planes can be written in the form $a\sqrt b+c$, where $b$ is a square-free positive integer.  Find $a+b+c$.
Answer: 432
Problem: How many ways are there to color the edges of a hexagon orange and black if we assume that two hexagons are indistinguishable if one can be rotated into the other? Note that we are saying the colorings OOBBOB and BOBBOO are distinct; we ignore flips.
Answer: 14
Problem: How many tuples of integers $(a_0,a_1,a_2,a_3,a_4)$ are there, with $1\leq a_i\leq 5$ for each $i$, so that $a_0<a_1>a_2<a_3>a_4$?
Answer: 246
Problem: A palindrome number is a positive integer that reads the same forward and backward. For example, $1221$ and $8$ are palindrome numbers whereas $69$ and $157$ are not.  $A$ and $B$ are $4$-digit palindrome numbers.  $C$ is a $3$-digit palindrome number.  Given that $A-B=C$, what is the value of $C$?
Answer: 121
Problem: Find the number of primes $p$ between $100$ and $200$ for which $x^{11}+y^{16}\equiv 2013\pmod p$ has a solution in integers $x$ and $y$.
Answer: 21
Problem: An integer sequence $a_1,a_2,\ldots,a_n$ has $a_1=0$, $a_n\leq 10$ and $a_{i+1}-a_i\geq 2$ for $1\leq i<n$.  How many possibilities are there for this sequence?  The sequence may be of any length.
Answer: 89
Problem: Triangle $A_1B_1C_1$ is an equilateral triangle with sidelength $1$.  For each $n>1$, we construct triangle $A_nB_nC_n$ from $A_{n-1}B_{n-1}C_{n-1}$ according to the following rule: $A_n,B_n,C_n$ are points on segments $A_{n-1}B_{n-1},B_{n-1}C_{n-1},C_{n-1}A_{n-1}$ respectively, and satisfy the following: \[\dfrac{A_{n-1}A_n}{A_nB_{n-1}}=\dfrac{B_{n-1}B_n}{B_nC_{n-1}}=\dfrac{C_{n-1}C_n}{C_nA_{n-1}}=\dfrac1{n-1}\]  So for example, $A_2B_2C_2$ is formed by taking the midpoints of the sides of $A_1B_1C_1$.  Now, we can write $\tfrac{|A_5B_5C_5|}{|A_1B_1C_1|}=\tfrac mn$ where $m$ and $n$ are relatively prime integers.  Find $m+n$.  (For a triangle $\triangle ABC$, $|ABC|$ denotes its area.)
Answer: 4891
Problem: If $x,y$ are real, then the $\textit{absolute value}$ of the complex number $z=x+yi$ is \[|z|=\sqrt{x^2+y^2}.\]  Find the number of polynomials $f(t)=A_0+A_1t+A_2t^2+A_3t^3+t^4$ such that $A_0,\ldots,A_3$ are integers and all roots of $f$ in the complex plane have absolute value $\leq 1$.
Answer: 43
Problem: Three chords of a sphere, each having length $5,6,7$, intersect at a single point inside the sphere and are pairwise perpendicular.  For $R$ the maximum possible radius of this sphere, find $R^2$.
Answer: 15
Problem: Let $O$ be a point with three other points $A,B,C$ and $\angle AOB=\angle BOC=\angle AOC=2\pi/3$.  Consider the average area of the set of triangles $ABC$ where $OA,OB,OC\in\{3,4,5\}$.  The average area can be written in the form $m\sqrt n$ where $m,n$ are integers and $n$ is not divisible by a perfect square greater than $1$.  Find $m+n$.
Answer: 15
Problem: Find the total number of triples of integers $(x,y,n)$ satisfying the equation $\tfrac 1x+\tfrac 1y=\tfrac1{n^2}$, where $n$ is either $2012$ or $2013$.
Answer: 338
Problem: Let $p, u, m, a, c$ be positive real numbers satisfying $5p^5+4u^5+3m^5+2a^5+c^5=91$. What is the
maximum possible value of:
\[18pumac + 2(2 + p)^2 + 23(1 + ua)^2 + 15(3 + mc)^2?\]
Answer: 873
Problem: Let $S$ be the set of integer triplets $(a, b, c)$ with $1 \le a \le b \le c$ that satisfy $a + b + c = 77$ and:
\[\frac{1}{a} +\frac{1}{b}+\frac{1}{c}= \frac{1}{5}.\]What is the value of the sum $\sum_{a,b,c  \in S} a\cdot b \cdot c$?
Answer: 15230
Problem: Consider a random permutation of the set $\{1, 2, . . . , 2015\}$. In other words, for each $1 \le i \le 2015$, $i$ is sent to the element $a_i$ where $a_i \in \{1, 2, . . . , 2015\}$ and if $i \neq j$, then $a_i \neq a_j$. What is the expected number of ordered pairs $(a_i, a_j )$ with $i - j > 155$ and $a_i - a_j > 266$?
Answer: 651222
Problem: Jonathan has a magical coin machine which takes coins in amounts of $7, 8$, and $9$. If he puts in $7$ coins, he gets $3$ coins back; if he puts in $8$, he gets $11$ back; and if he puts in $9$, he gets $4$ back. The coin machine does not allow two entries of the same amount to happen consecutively. Starting with $15$ coins, what is the minimum number of entries he can make to end up with $4$ coins?
Answer: 4
Problem: In how many ways can $9$ cells of a $\text{6-by-6}$ grid be painted black such that no two black cells share a corner or an edge with each other?
Answer: 3600
Problem: Define the sequence $a_i$ as follows: $a_1 = 1, a_2 = 2015$, and $a_n = \frac{na_{n-1}^2}{a_{n-1}+na_{n-2}}$ for $n > 2$. What is the least $k$ such that $a_k < a_{k-1}$?
Answer: 7
Problem: Charlie noticed his golden ticket was golden in two ways! In addition to being gold, it was a rectangle whose side lengths had ratio the golden ratio $\varphi = \tfrac{1+\sqrt{5}}{2}$. He then folds the ticket so that two opposite corners (vertices connected by a diagonal) coincide and makes a sharp crease (the ticket folds just as any regular piece of paper would). The area of the resulting shape can be expressed as $a + b \varphi$. What is $\tfrac{b}{a}$?
Answer: 2
Problem: Find the largest $r$ such that $4$ balls each of radius $r$ can be packed into a regular tetrahedron with side length $1$. In a packing, each ball lies outside every other ball, and every ball lies inside the boundaries of the tetrahedron. If $r$ can be expressed in the form $\frac{\sqrt{a}+b}{c}$ where $a, b, c$ are integers such that $\gcd(b, c) = 1$, what is $a + b + c$?
Answer: 15
Problem: A number is [i]interesting [/i]if it is a $6$-digit integer that contains no zeros, its first $3$ digits are strictly increasing, and its last $3$ digits are non-increasing. What is the average of all interesting numbers?
Answer: 308253
Problem: What is the largest positive integer $n$ less than $10, 000$ such that in base $4$, $n$ and $3n$ have the same number of digits; in base $8$, $n$ and $7n$ have the same number of digits; and in base $16$, $n$ and $15n$ have the same number of digits? Express your answer in base $10$.
Answer: 4369
Problem: A word is an ordered, non-empty sequence of letters, such as $word$ or $wrod$. How many distinct $3$-letter words can be made from a subset of the letters $c, o, m, b, o$, where each letter in the list is used no more than the number of times it appears?
Answer: 170
Problem: We define the function $f(x,y)=x^3+(y-4)x^2+(y^2-4y+4)x+(y^3-4y^2+4y)$. Then choose any distinct $a, b, c \in \mathbb{R}$ such that the following holds: $f(a,b)=f(b,c)=f(c,a)$. Over all such choices of $a, b, c$, what is the maximum value achieved by
\[\min(a^4 - 4a^3 + 4a^2, b^4 - 4b^3 + 4b^2, c^4 - 4c^3 + 4c^2)?\]
Answer: 1
Problem: For a positive integer $n$, let $d(n)$ be the number of positive divisors of $n$. What is the smallest positive integer $n$ such that
\[\sum_{t \mid n} d(t)^3\]is divisible by $35$?
Answer: 864
Problem: Triangle $ABC$ has $\overline{AB} = 5, \overline{BC} = 4, \overline{CA} = 6$. Points $D$ and $E$ are on sides $AB$ and $AC$, respectively, such that $\overline{AD} = \overline{AE} = \overline{BC}$. Let $CD$ and $BE$ intersect at $F$ and let $AF$ and $DE$ intersect at $G$. The length of $FG$ can be expressed in the form $\tfrac{a\sqrt{b}}{c}$ in simplified form. What is $a + b + c$?
Answer: 255
Problem: The incircle of acute triangle $ABC$ touches $BC, AC$, and $AB$ at points $D, E$, and $F$, respectively. Let $P$ be the second intersection of line $AD$ and the incircle. The line through $P$ tangent to the incircle intersects $AB$ and $AC$ at points $M$ and $N$, respectively. Given that $\overline{AB} = 8, \overline{AC} = 10$, and $\overline{AN} = 4$, let $\overline{AM} = \tfrac{a}{b}$ where $a$ and $b$ are positive coprime integers. What is $a + b$?
Answer: 49
Problem: We define the ridiculous numbers recursively as follows:
[list=a]
[*]1 is a ridiculous number.
[*]If $a$ is a ridiculous number, then $\sqrt{a}$ and $1+\sqrt{a}$ are also ridiculous numbers. 
[/list]
A closed interval $I$ is ``boring'' if
[list]
[*]$I$ contains no ridiculous numbers, and
[*]There exists an interval $[b,c]$ containing $I$ for which $b$ and $c$ are both ridiculous numbers.
[/list]
The smallest non-negative $l$ such that there does not exist a boring interval with length $l$ can be represented in the form $\dfrac{a + b\sqrt{c}}{d}$ where $a, b, c, d$ are integers, $\gcd(a, b, d) = 1$, and no integer square greater than 1 divides $c$. What is $a + b + c + d$?
Answer: 9
Problem: Triangle $ABC$ has $\overline{AB} = \overline{AC} = 20$ and $\overline{BC} = 15$. Let $D$ be the point in $\triangle ABC$ such that $\triangle ADB \sim \triangle BDC$. Let $l$ be a line through $A$ and let $BD$ and $CD$ intersect $l$ at $P$ and $Q$, respectively. Let the circumcircles of $\triangle BDQ$ and $\triangle CDP$ intersect at $X$. The area of the locus of $X$ as $l$ varies can be expressed in the form $\tfrac{p}{q}\pi$ for positive coprime integers $p$ and $q$. What is $p + q$?
Answer: 1291
Problem: Terry the Tiger lives on a cube-shaped world with edge length $2$. Thus he walks on the outer surface. He is tied, with a leash of length $2$, to a post located at the center of one of the faces of the cube. The surface area of the region that Terry can roam on the cube can be represented as $\frac{p \pi}{q} + a\sqrt{b}+c$ for integers $a, b, c, p, q$ where no integer square greater than $1$ divides $b, p$ and $q$ are coprime, and $q > 0$. What is $p + q + a + b + c$? (Terry can be at a location if the shortest distance along the surface of the cube between that point and the post is less than or equal to $2$.)
Answer: 14
Problem: What is the $22\text{nd}$ positive integer $n$ such that $22^n$ ends in a $2$? (when written in base $10$).
Answer: 85
Problem: Andrew and Blair are bored in class and decide to play a game. They pick a pair $(a, b)$ with $1 \le a, b \le 100$. Andrew says the next number in the geometric series that begins with $a,b$ and Blair says the next number in the arithmetic series that begins with $a,b$. For how many pairs $(a, b)$ is Andrew's number minus Blair's number a positive perfect square?
Answer: 281
Problem: For her daughter’s $12\text{th}$ birthday, Ingrid decides to bake a dodecagon pie in celebration. Unfortunately, the store does not sell dodecagon shaped pie pans, so Ingrid bakes a circular pie first and then trims off the sides in a way such that she gets the largest regular dodecagon possible. If the original pie was $8$ inches in diameter, the area of pie that she has to trim off can be represented in square inches as $a\pi - b$ where $a, b$ are integers. What is $a + b$?
Answer: 64
Problem: Given a rational number $r$ that, when expressed in base-$10$, is a repeating, non-terminating decimal, we define $f(r)$ to be the number of digits in the decimal representation of $r$ that are after the decimal point but before the repeating part of $r$. For example, $f(1.2\overline{7}) = 0$ and $f(0.35\overline{2}) = 2$. What is the smallest positive integer $n$ such that $\tfrac{1}{n}, \tfrac{2}{n}$, and $\tfrac{4}{n}$ are non-terminating decimals, where $f\left( \tfrac{1}{n} \right) = 3, f\left( \tfrac{2}{n} \right) = 3$, and $f\left( \tfrac{4}{n} \right) = 2$?.
Answer: 600
Problem: Given that there are $24$ primes between $3$ and $100$, inclusive, what is the number of ordered pairs $(p, a)$ with $p$ prime, $3 \le p < 100$, and $1 \le a < p$ such that $p \mid (a^{p-2} - a)$?
Answer: 48
Problem: A circle with radius $1$ and center $(0, 1)$ lies on the coordinate plane. Ariel stands at the origin and rolls a ball of paint at an angle of $35$ degrees relative to the positive $x$-axis (counting degrees counterclockwise). The ball repeatedly bounces off the circle and leaves behind a trail of paint where it rolled. After the ball of paint returns to the origin, the paint has traced out a star with $n$ points on the circle. What is $n$?
Answer: 36
Problem: Find the sum of the non-repeated roots of the polynomial $P(x) = x^6-5x^5-4x^4-5x^3+8x^2+7x+7$.
Answer: 7