sungyub/dapo-math-17k-verl huggingface dataset:
user: Regions $A, B, C, J$ and $K$ represent ponds. Logs leave pond $A$ and float down flumes (represented by arrows) to eventually end up in pond $B$ or pond $C$. On leaving a pond, the logs are equally likely to use any available exit flume. Logs can only float in the direction the arrow is pointing. What is the probability that a log in pond $A$ will end up in pond $B$? Express your answer as a common fraction.

label("$A$",(10,22),S);
label("$B$",(10,2),S);
label("$C$",(10,-7),S);
label("$J$",(2,13),S);
label("$K$",(10,12),S);
path a=(10,-10)..(20,0)--(0,0)..cycle;
path b=(10,-7)..(3,0)--(17,0)..cycle;
draw(a);
draw(b);
fill((3.1,0.1)--(16.9,0.1)--(16.9,-0.1)--(3.1,-0.1)--cycle,white);
draw(Circle((10,0),3));
draw(Circle((10,10),3));
draw(Circle((10,20),3));
draw((10,16.9)--(10,13.1),Arrow);
draw((10,6.9)--(10,3.1),Arrow);
draw(Circle((2,10),3));
draw((2,7)--(2.5,0),Arrow);
draw((1.5,7)--(0.2,0),Arrow);
draw((10,16.9)--(2,13.2),Arrow);
draw((10,16.9)--(19.8,0),Arrow);
draw((10,6.9)--(17.2,0),Arrow);
draw((3,8)--(10,3.1),Arrow);
[/asy]The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.

Answer: 23
user: Rectangle $ABCD$ is inscribed in triangle $EFG$ such that side $AD$ of the rectangle is on side $EG$ of the triangle, as shown. The triangle's altitude from $F$ to side $EG$ is 7 inches, and $EG = 10 \text{ inches}$. The length of segment $AB$ is equal to half the length of segment $AD$. What is the area of rectangle $ABCD$? Express your answer as a common fraction.

import math;
size(101);
real x = 35/12;
currentpen = linewidth(1)+fontsize(10pt);
pair E1 = (0,0), G = (10,0), F = (3,7), A = (3*x/7,0), D = G - (7*x/7,0), B = extension(E1,F,A,A+(0,1)), C = extension(G,F,D,D+(0,1));
draw(E1--F--G--cycle); draw(A--B--C--D); label("$A$",A,S); label("$B$",B,NW); label("$C$",C,NE); label("$D$",D,S); label("$E$",E1,W); label("$F$",F,NNW); label("$G$",G,ESE);
[/asy]The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.

Answer: 1297
user: Circles with centers at $O$ and $P$ have radii 2 and 4, respectively, and are externally tangent. Points $A$ and $B$ on the circle with center $O$ and points $C$ and $D$ on the circle with center $P$ are such that $\overline{AD}$ and $\overline{BC}$ are common external tangents to the circles. What is the area of the concave hexagon $AOBCPD$?

pair A,B,C,D,H,O,P;
O = (0,0);
P = (6,0);
H = (-6,0);
A = intersectionpoint(arc((A + H)/2, abs(A - H)/2, 0, 180), arc(O, 2, 0, 180));
D = intersectionpoint(arc((A + P)/2, abs(A - P)/2, 0, 180), arc(P, 4, 0, 180));
B = reflect(O,P)*(A);
C = reflect(O,P)*(D);
draw(O--P,linewidth(0.7));
draw(Circle(O,2),linewidth(0.7));
draw(Circle(P,4),linewidth(0.7));
draw(interp(A,D,-0.2)--interp(A,D,1.2),linewidth(0.7));
draw(interp(B,C,-0.2)--interp(B,C,1.2),linewidth(0.7));
draw(A--O--B,linewidth(0.7));
draw(D--P--C,linewidth(0.7));
label("$O$",O,SE);
label("$P$",P,E);
label("$D$",D,NW);
label("$A$",A,NW);
label("$B$",B,S);
label("$C$",C,S);
label("2",(-0.45,0.95),E);
label("6",(3,0),N);
label("4",(5.55,1.95),E);

[/asy]The answer is in the form k\sqrt{m}+n,. Please provide the value of k + m + n.

Answer: 26
user: Tina randomly selects two distinct numbers from the set $$\{1,2,3,4,5\},$$and Sergio randomly selects a number from the set $$\{1,2,\ldots,10\}.$$What is the probability that Sergio's number is larger than the sum of the two numbers chosen by Tina?The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.

Answer: 7
user: What is the value of $x$ in the diagram?

import olympiad;

draw((0,0)--(sqrt(3),0)--(0,sqrt(3))--cycle);

draw((0,0)--(-3,0)--(0,sqrt(3))--cycle);

label("$2\sqrt{3}$",(-3/2,sqrt(3)/2),NW);

label("$x$",(sqrt(3)/2,sqrt(3)/2),NE);

draw("$45^{\circ}$",(1.5,0),NW);

draw("$30^{\circ}$",(-2.45,0),NE);

draw(rightanglemark((0,sqrt(3)),(0,0),(sqrt(3),0),5));

[/asy]The answer is in the form k\sqrt{m}+n,. Please provide the value of k + m + n.

Answer: 7
user: Points $A$, $B$, $C$, and $T$ are in space such that each of $\overline{TA}$, $\overline{TB}$, and $\overline{TC}$ is perpendicular to the other two. If $TA = TB = 12$ and $TC = 6$, then what is the distance from $T$ to face $ABC$?The answer is in the form k\sqrt{m}+n,. Please provide the value of k + m + n.

Answer: 8
user: For $1 \leq i \leq 215$ let $a_i = \dfrac{1}{2^{i}}$ and $a_{216} = \dfrac{1}{2^{215}}$. Let $x_1, x_2, \dots, x_{216}$ be positive real numbers such that $\sum_{i=1}^{216} x_i=1$ and
\[\sum_{1 \leq i < j \leq 216} x_ix_j = \dfrac{107}{215} + \sum_{i=1}^{216} \dfrac{a_i x_i^{2}}{2(1-a_i)}.\]Find the maximum possible value of $x_2.$The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.

Answer: 863
user: Compute
\[\cos \frac{2 \pi}{7} \cos \frac{4 \pi}{7} \cos \frac{8 \pi}{7}.\]The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.

Answer: 9
user: A binary operation $\diamondsuit$ has the properties that $a\,\diamondsuit\, (b\,\diamondsuit \,c) = (a\,\diamondsuit \,b)\cdot c$ and that $a\,\diamondsuit \,a=1$ for all nonzero real numbers $a, b,$ and $c$. (Here $\cdot$ represents multiplication). Find the solution to the equation $2016 \,\diamondsuit\, (6\,\diamondsuit\, x)=100.$The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.

Answer: 109
user: Find the area of triangle $JKL$ below.

unitsize(1inch);
pair P,Q,R;
P = (0,0);
Q= (sqrt(3),0);
R = (0,1);
draw (P--Q--R--P,linewidth(0.9));
draw(rightanglemark(Q,P,R,3));
label("$J$",P,S);
label("$K$",Q,S);
label("$L$",R,N);
label("$20$",(Q+R)/2,NE);
label("$60^\circ$",(0,0.75),E);
[/asy]The answer is in the form k\sqrt{m}+n,. Please provide the value of k + m + n.

Answer: 53
user: If
\[x + \sqrt{x^2 - 1} + \frac{1}{x - \sqrt{x^2 - 1}} = 20,\]then find
\[x^2 + \sqrt{x^4 - 1} + \frac{1}{x^2 + \sqrt{x^4 - 1}}.\]The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.

Answer: 10401
user: Let $a,$ $b,$ and $c$ be positive real numbers such that $a + b^2 + c^3 = \frac{325}{9}.$ Find the minimum value of
\[a^2 + b^3 + c^4.\]The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.

Answer: 2834
user: Suppose we flip four coins simultaneously: a penny, a nickel, a dime, and a quarter. What is the probability that at least 15 cents worth of coins come up heads?The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.

Answer: 13
user: For positive real numbers $x,$ $y,$ and $z,$ compute the maximum value of
\[\frac{xyz(x + y + z)}{(x + y)^2 (y + z)^2}.\]The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.

Answer: 5
user: An integer is randomly chosen from the integers 1 through 100, inclusive. What is the probability that the chosen integer is a perfect square or a perfect cube, but not both? Express your answer as a common fraction.The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.

Answer: 11
user: Let $x,$ $y,$ and $z$ be positive real numbers. Find the maximum value of
\[\frac{xyz}{(1 + 5x)(4x + 3y)(5y + 6z)(z + 18)}.\]The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.

Answer: 5121
user: Let $x$ and $y$ be real numbers such that $x + y = 3.$ Find the maximum value of
\[x^4 y + x^3 y + x^2 y + xy + xy^2 + xy^3 + xy^4.\]The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.

Answer: 411
user: A right pyramid with a square base has total surface area 432 square units. The area of each triangular face is half the area of the square face. What is the volume of the pyramid in cubic units?The answer is in the form k\sqrt{m}+n,. Please provide the value of k + m + n.

Answer: 291
user: All the complex roots of $(z + 1)^5 = 32z^5,$ when plotted in the complex plane, lie on a circle. Find the radius of this circle.The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.

Answer: 5
user: When three standard dice are tossed, the numbers $a,b,c$ are obtained. Find the probability that $$(a-1)(b-1)(c-1) \neq 0$$The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.

Answer: 341
user: An envelope contains eight bills: 2 ones, 2 fives, 2 tens, and 2 twenties. Two bills are drawn at random without replacement. What is the probability that their sum is $\$20$ or more?The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.

Answer: 3
user: Find the distance between the vertices of the hyperbola $9x^2 + 54x - y^2 + 10y + 55 = 0.$The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.

Answer: 5
user: Let $(x,y)$ be a pair of real numbers satisfying \[
56x + 33y = \frac{-y}{x^2+y^2}, \qquad \text{and} \qquad 33x-56y = \frac{x}{x^2+y^2}.
\]Determine the value of $|x| + |y|$.The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.

Answer: 76
user: A regular hexagon of side length 1 is placed vertically along one of its edges on a flat surface. A particle is fired from the surface and passes through four of the vertices of the hexagon, as shown below, tracing the arc of a parabola. If the particle starts at $A$ and ends at $B,$ then find the length $AB.$

unitsize(1.5 cm);

real func (real x) {
 return(-2/sqrt(3)*x^2 + 7/(2*sqrt(3)));
}

pair A, B;

A = (-sqrt(7)/2,0);
B = (sqrt(7)/2,0);

draw(shift((0,sqrt(3)/2))*(dir(240)--dir(180)--dir(120)--dir(60)--dir(0)--dir(-60)));
draw((-2,0)--(2,0));
draw(graph(func,-sqrt(7)/2,sqrt(7)/2),red);

label("$A$", A, S);
label("$B$", B, S);
[/asy]The answer is in the form k\sqrt{m}+n,. Please provide the value of k + m + n.

Answer: 8
user: In triangle $ABC$,
\[2a^2 + 4b^2 + c^2 = 4ab + 2ac.\]Compute the numerical value of $\cos B.$The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.

Answer: 15
user: What is the average (mean) of all $5$-digit numbers that can be formed by using each of the digits $1$, $3$, $5$, $7$, and $8$ exactly once?The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.

Answer: 266669
user: Three of the four endpoints of the axes of an ellipse are, in some order, \[(-2, 4), \; (3, -2), \; (8, 4).\]Find the distance between the foci of the ellipse.The answer is in the form k\sqrt{m}+n,. Please provide the value of k + m + n.

Answer: 13
user: Find the number of units in the length of diagonal $DA$ of the regular hexagon shown. Express your answer in simplest radical form. [asy]
size(120);
draw((1,0)--(3,0)--(4,1.732)--(3,3.464)--(1,3.464)--(0,1.732)--cycle);
draw((1,0)--(1,3.464));
label("10",(3.5,2.598),NE);
label("$A$",(1,0),SW);
label("$D$",(1,3.464),NW);
[/asy]The answer is in the form k\sqrt{m}+n,. Please provide the value of k + m + n.

Answer: 13
user: If two distinct members of the set $\{ 2, 4, 12, 14, 21, 28, 98 \}$ are randomly selected and multiplied, what is the probability that the product is a multiple of 196? Express your answer as a common fraction.The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.

Answer: 4
user: In triangle $ABC,$ $D,$ $E,$ and $F$ are points on sides $\overline{BC},$ $\overline{AC},$ and $\overline{AB},$ respectively, so that $BD:DC = CE:EA = AF:FB = 1:2.$

unitsize(0.8 cm);

pair A, B, C, D, E, F, P, Q, R;

A = (2,5);
B = (0,0);
C = (7,0);
D = interp(B,C,1/3);
E = interp(C,A,1/3);
F = interp(A,B,1/3);
P = extension(A,D,C,F);
Q = extension(A,D,B,E);
R = extension(B,E,C,F);

fill(P--Q--R--cycle,gray(0.7));
draw(A--B--C--cycle);
draw(A--D);
draw(B--E);
draw(C--F);

label("$A$", A, N);
label("$B$", B, SW);
label("$C$", C, SE);
label("$D$", D, S);
label("$E$", E, NE);
label("$F$", F, W);
label("$P$", P, NE);
label("$Q$", Q, NW);
label("$R$", R, S);
[/asy]

Line segments $\overline{AD},$ $\overline{BE},$ and $\overline{CF}$ intersect at $P,$ $Q,$ and $R,$ as shown above. Compute $\frac{[PQR]}{[ABC]}.$The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.

Answer: 8
user: The line $x = k$ intersects the graph of the parabola $x = -2y^2 - 3y + 5$ at exactly one point. What is $k$?The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.

Answer: 57
user: Spinners $A$ and $B$ are spun. On each spinner, the arrow is equally likely to land on each number. What is the probability that the product of the two spinners' numbers is even? Express your answer as a common fraction. [asy]
/* AMC8 2004 #21 Problem */
draw(circle((0,0), 2));
draw(circle((6,0), 2));
draw((-2,0)--(2,0));
draw((0,-2)--(0,2));
path p=(0,0)--(0,2);
draw(shift(6,0)*p);
draw(shift(6,0)*rotate(120)*p);
draw(shift(6,0)*rotate(240)*p);
label(scale(0.6)*"1", (-1,1));
label(scale(0.6)*"2", (1,1));
label(scale(0.6)*"3", (1,-1));
label(scale(0.6)*"4", (-1,-1));
label(scale(0.6)*"1", (5,.75));
label(scale(0.6)*"2", (7,.75));
label(scale(0.6)*"3", (6,-1));
draw((0,0)--(1.5,.5), EndArrow);
draw(shift(6,0)*rotate(45)*((0,0)--(1.5,.5)), EndArrow);
[/asy]The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.

Answer: 5
user: Solve for $x$: $\sqrt[3]{20x + \sqrt[3]{20x + 13}} = 13.$The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.

Answer: 551
user: Find the maximum value of
\[\cos \theta_1 \sin \theta_2 + \cos \theta_2 \sin \theta_3 + \cos \theta_3 \sin \theta_4 + \cos \theta_4 \sin \theta_5 + \cos \theta_5 \sin \theta_1,\]over all real numbers $\theta_1,$ $\theta_2,$ $\theta_3,$ $\theta_4,$ and $\theta_5.$The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.

Answer: 7
user: Let $w$ and $z$ be complex numbers such that $|w| = 1$ and $|z| = 10$. Let $\theta = \arg \left(\frac{w-z}{z}\right)$. Find the maximum possible value of $\tan^2 \theta.$ (Note that $\arg(w),$ for $w \neq 0,$ denotes the measure of the angle that the ray from 0 to $w$ makes with the positive real axis in the complex plane.)The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.

Answer: 100
user: Let the roots of
\[z^3 = 2 + 2i\]be $a_1 + ib_1,$ $a_2 + ib_2,$ and $a_3 + ib_3.$ Compute $a_1 a_2 a_3.$The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.

Answer: 3
user: In the coordinate plane, the graph of
\[|x + y - 1| + \Big| |x| - x \Big| + \Big| |x - 1| + x - 1 \Big| = 0\]is a certain curve. Find the length of this curve.The answer is in the form k\sqrt{m}+n,. Please provide the value of k + m + n.

Answer: 3
user: The chart below gives the air distance in miles between selected world cities. If two different cities from the chart are chosen at random, what is the probability that the distance between them is less than $7000$ miles? Express your answer as a common fraction.

\begin{tabular}{|c|c|c|c|c|}
\hline
& Bangkok & Cape Town & Honolulu & London \\ \hline
Bangkok & & 6300 & 6609 & 5944 \\ \hline
Cape Town & 6300 & & 11,535 & 5989 \\ \hline
Honolulu & 6609 & 11,535 & & 7240 \\ \hline
London & 5944 & 5989 & 7240 & \\ \hline
\end{tabular}The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.

Answer: 5
user: The altitude to the hypotenuse of a triangle with angles of 30 and 60 degrees is 3 units. What is the area of the triangle, in square units? Express your answer in simplest radical form. [asy]
unitsize(6mm);
defaultpen(linewidth(.7pt)+fontsize(8pt));

real r=2*sqrt(3);

pair A=r*dir(0), B=r*dir(60), C=r*dir(180);
pair F=foot(B,A,C);
draw(A--B--C--cycle);
draw(rightanglemark(A,B,C,8));
draw(B--F,linetype("4 2"));
label("3 units",waypoint(B--F,0.6),W);
[/asy]The answer is in the form k\sqrt{m}+n,. Please provide the value of k + m + n.

Answer: 9
user: Evaluate $\frac{7!+8!}{6!+7!}$. Express your answer as a common fraction.The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.

Answer: 71
user: $ABCD$ is a regular tetrahedron (right triangular pyramid). If $M$ is the midpoint of $\overline{CD}$, then what is $\tan\angle AMB$?The answer is in the form k\sqrt{m}+n,. Please provide the value of k + m + n.

Answer: 4
user: Four prime numbers are randomly selected without replacement from the first ten prime numbers. What is the probability that the sum of the four selected numbers is odd? Express your answer as a common fraction.The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.

Answer: 7
user: Five points $A$, $B$, $C$, $D$, and $O$ lie on a flat field. $A$ is directly north of $O$, $B$ is directly west of $O$, $C$ is directly south of $O$, and $D$ is directly east of $O$. The distance between $C$ and $D$ is 140 m. A hot-air balloon is positioned in the air at $H$ directly above $O$. The balloon is held in place by four ropes $HA$, $HB$, $HC$, and $HD$. Rope $HC$ has length 150 m and rope $HD$ has length 130 m. How high is the balloon above the field (that is, the length of $OH$)? [asy]
size(200);
pair A, B, C, D, O, H, W, X, Y, Z;
O=(0,0);
A=(1,1);
D=(1.5,-.3);
B=(-1.5,.3);
C=(-1,-1);
H=(0,2.5);
W=(5/3)*(A+D);
X=(5/3)*(A+B);
Y=(-1)*(W);
Z=(-1)*(X);
draw(W--X--Y--Z--W);
draw(A--C);
draw(B--D);
draw(O--H, linewidth(1));
draw(A--H, dashed);
draw(B--H, dashed);
draw(C--H, dashed);
draw(D--H, dashed);
dot(A);
dot(B);
dot(C);
dot(D);
dot(O);
dot(H);
label("A", A, NE);
label("B", B, SW);
label("C", C, SE);
label("D", D, NE);
label("O", O, SE);
label("H", H, NW);
[/asy]The answer is in the form k\sqrt{m}+n,. Please provide the value of k + m + n.

Answer: 41
user: Let $a,$ $b,$ $c,$ $d$ be real numbers such that $a + b + c + d = 17$ and $ab + bc + cd + da = 46.$ Find the minimum possible value of $a^2 + b^2 + c^2 + d^2.$The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.

Answer: 199
user: What is the value of the sum
\[
 \sum_z \frac{1}{{\left|1 - z\right|}^2} \, ,
\]where $z$ ranges over all 7 solutions (real and nonreal) of the equation $z^7 = -1$?The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.

Answer: 53
user: A point $P$ is randomly placed in the interior of the right triangle below. What is the probability that the area of triangle $PBC$ is less than half of the area of triangle $ABC$? Express your answer as a common fraction. [asy]
size(7cm);
defaultpen(linewidth(0.7));
pair A=(0,5), B=(8,0), C=(0,0), P=(1.5,1.7);
draw(A--B--C--cycle);
draw(C--P--B);
label("$A$",A,NW);
label("$B$",B,E);
label("$C$",C,SW);
label("$P$",P,N);
draw((0,0.4)--(0.4,0.4)--(0.4,0));[/asy]The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.

Answer: 7
user: Find the minimum value of
\[\sqrt{x^2 + (1 - x)^2} + \sqrt{(1 - x)^2 + (1 + x)^2}\]over all real numbers $x.$The answer is in the form k\sqrt{m}+n,. Please provide the value of k + m + n.

Answer: 6
user: In the game of Twister, a spinner randomly selects either an arm or a leg, and also selects one of four colors, one of which is red, each with equal probability, and players have to move the appropriate body part to the appropriately colored spot on the ground. There are four players. Each player spins once, and makes the move the spinner selects. What is the probability that in these four spins, there will be exactly two moves to a red spot, and the body part selected to move will be an arm exactly $3$ times?The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.

Answer: 539
user: 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 $\overline{AD}$? Express your answer as a common fraction.

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]The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.

Answer: 21
user: Suppose
\[\frac{1}{x^3-x^2-21x+45}=\frac{A}{x+5}+\frac{B}{x-3} + \frac{C}{(x - 3)^2}\]where $A$, $B$, and $C$ are real constants. What is $A$?The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.

Answer: 65
user: Michael has never taken a foreign language class, but is doing a story on them for the school newspaper. The school offers French and Spanish. Michael has a list of all 25 kids in the school enrolled in at least one foreign language class. He also knows that 18 kids are in the French class and 21 kids are in the Spanish class. If Michael chooses two kids at random off his list and interviews them, what is the probability that he will be able to write something about both the French and Spanish classes after he is finished with the interviews? Express your answer as a fraction in simplest form.The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.

Answer: 191
user: Express $0.5\overline{10}$ as a common fraction.The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.

Answer: 299
user: Joan tries to solve a really hard problem once each day. She has a 1/4 probability of solving it each day. What is the probability that she will solve it before her sixth try?The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.

Answer: 1805
user: Two distinct positive integers from 1 to 50 inclusive are chosen. Let the sum of the integers equal $S$ and the product equal $P$. What is the probability that $P+S$ is one less than a multiple of 5?The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.

Answer: 334
user: In triangle $ABC,$ $\angle B = 30^\circ,$ $AB = 150,$ and $AC = 50 \sqrt{3}.$ Find the sum of all possible values of $BC.$The answer is in the form k\sqrt{m}+n,. Please provide the value of k + m + n.

Answer: 153
user: Let $\mathcal{T}$ be the set of ordered triples $(x,y,z)$ of nonnegative real numbers that lie in the plane $x+y+z=1.$ Let us say that $(x,y,z)$ supports $(a,b,c)$ when exactly two of the following are true: $x\ge a, y\ge b, z\ge c.$ Let $\mathcal{S}$ consist of those triples in $\mathcal{T}$ that support $\left(\frac 12,\frac 13,\frac 16\right).$ Find the area of $\mathcal{S}$ divided by the area of $\mathcal{T}.$The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.

Answer: 25
user: Let $z_1,$ $z_2,$ $\dots,$ $z_{20}$ be the twenty (complex) roots of the equation
\[z^{20} - 4z^{19} + 9z^{18} - 16z^{17} + \dots + 441 = 0.\]Calculate $\cot \left( \sum_{k = 1}^{20} \operatorname{arccot} z_k \right).$ Note that the addition formula for cotangent is still valid when working with complex numbers.The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.

Answer: 461
user: John needs to catch a train. The train arrives randomly some time between 2:00 and 3:00, waits for 20 minutes, and then leaves. If John also arrives randomly between 2:00 and 3:00, what is the probability that the train will be there when John arrives?The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.

Answer: 23