sungyub/dapo-math-17k-verl huggingface dataset:
user: Given an integer $a_0$, we define a sequence of real numbers $a_0, a_1, . . .$ using the relation $$a^2_i = 1 + ia^2_{i-1},$$
for $i \ge 1$. An index $j$ is called [i]good [/i] if $a_j$ can be an integer for some $a_0$. Determine the sum of the indices $j$ which lie in the interval $[0, 99]$ and which are not good.

Answer: 4946
user: Jack plays a game in which he first rolls a fair six-sided die and gets some number $n$, then, he flips a coin until he flips $n$ heads in a row and wins, or he flips $n$ tails in a row in which case he rerolls the die and tries again. What is the expected number of times Jack must flip the coin before he wins the game.

Answer: 40
user: The smallest three positive proper divisors of an integer n are $d_1 < d_2 < d_3$ and they satisfy $d_1 + d_2 + d_3 = 57$. Find the sum of the possible values of $d_2$.

Answer: 42
user: Consider the following expression
$$S = \log_2 \left( \sum^{2019}_{k=1}\sum^{2020}_{j=2}\log_{2^{1/k}} (j) \log_{j^2} \left(\sin \frac{\pi k}{2020}\right) \right).$$
Find the smallest integer $n$ which is bigger than $S$ (i.e. find $\lceil S \rceil$).

Answer: 31
user: Abby and Ben have a little brother Carl who wants candy. Abby has $7$ different pieces of candy and Ben has $15$ different pieces of candy. Abby and Ben then decide to give Carl some candy. As Ben wants to be a better sibling than Abby, so he decides to give two more pieces of candy to Carl than Abby does. Let $N$ be the number of ways Abby and Ben can give Carl candy. Compute the number of positive divisors of $N$.

Answer: 96
user: Consider the sequence of Fibonacci numbers $F_0, F_1, F_2, ... $, given by $F_0 = F_1= 1$ and $F_{n+1} =F_n + F_{n-1}$ for $n\ge 1$. Define the sequence $x_0, x_1, x_2, ....$ by $x_0 = 1$ and $x_{k+1} = x^2_k + F^2_{2^k}$ for $k \ge 0$. Define the sequence $y_0, y_1, y_2, ...$ by $y_0 = 1$ and $y_{k+1} = 2x_ky_k - y^2_k$ for $k \ge 0$. If $$\sum^{\infty}_{k=0} \frac{1}{y_k}= \frac{a -\sqrt{b}}{c}$$
for positive integers a$, b, c$ with $gcd (a, c) = 1$, find $a + b + c$.

Answer: 14
user: The new PUMaC tournament hosts $2020$ students, numbered by the following set of labels $1, 2, . . . , 2020$. The students are initially divided up into $20$ groups of $101$, with each division into groups equally likely. In each of the groups, the contestant with the lowest label wins, and the winners advance to the second round. Out of these $20$ students, we chose the champion uniformly at random. If the expected value of champion’s number can be written as $\frac{a}{b}$, where $a, b$ are relatively prime integers, determine $a + b$.

Answer: 2123
user: Physicists at Princeton are trying to analyze atom entanglement using the following experiment. Originally there is one atom in the space and it starts splitting according to the following procedure. If after $n$ minutes there are atoms $a_1, \dots, a_N$, in the following minute every atom $a_i$ splits into four new atoms, $a_i^{(1)},a_i^{(2)},a_i^{(3)},a_i^{(4)}$. Atoms $a_i^{(j)}$ and $a_k^{(j)}$ are entangled if and only the atoms $a_i$ and $a_k$ were entangled after $n$ minutes. Moreover, atoms $a_i^{(j)}$ and $a_k^{(j+1)}$ are entangled for all $1 \le i$, $k \le N$ and $j = 1$, $2$, $3$. Therefore, after one minute there is $4$ atoms, after two minutes there are $16$ atoms and so on.

Physicists are now interested in the number of unordered quadruplets of atoms $\{b_1, b_2, b_3, b_4\}$ among which there is an odd number of entanglements. What is the number of such quadruplets after $3$ minutes?

Remark[/i]. Note that atom entanglement is not transitive. In other words, if atoms $a_i$, $a_j$ are entangled and if $a_j$, $a_k$ are entangled, this does not necessarily mean that $a_i$ and $a_k$ are entangled.

Answer: 354476
user: Alice, Bob, and Carol are playing a game. Each turn, one of them says one of the $3$ players' names, chosen from {Alice, Bob, Carol} uniformly at random. Alice goes first, Bob goes second, Carol goes third, and they repeat in that order. Let $E$ be the expected number of names that are have been said when, for the first time, all $3$ names have been said twice. If $E = \tfrac{m}{n}$ for relatively prime positive integers $m$ and $n$, find $m + n$. (Include the last name to be said twice in your count.)

Answer: 383
user: Cassidy has string of $n$ bits, where $n$ is a positive integer, which initially are all $0$s or $1$s. Every second, Cassidy may choose to do one of two things:

Change the first bit (so the first bit changes from a $0$ to a $1$, or vice versa)
Change the first bit after the first $1$.

Let $M$ be the minimum number of such moves it takes to get from $1\dots 1$ to $0 \dots 0$ (both of length $12$), and $N$ the number of starting sequences with $12$ bits that Cassidy can turn into all $0$s. Find $M + N$.

Answer: 6826
user: Let $AX$ be a diameter of a circle $\Omega$ with radius $10$, and suppose that $C$ lies on $\Omega$ so that $AC = 16$. Let $D$ be the other point on $\Omega$ so $CX = CD$. From here, define $D'$ to be the reflection of $D$ across the midpoint of $AC$, and $X'$ to be the reflection of $X$ across the midpoint of $CD$. If the area of triangle $CD'X'$ can be written as $\frac{p}{q}$ , where $p, q$ are relatively prime, find $p + q$.

Answer: 1367
user: Let $f(x) = 1 + 2x + 3x^2 + 4x^3 + 5x^4$ and let $\zeta = e^{2\pi i/5} = \cos \frac{2\pi}{5} + i \sin \frac{2\pi}{5}$. Find the value of the following expression: $$f(\zeta)f(\zeta^2)f(\zeta^3)f(\zeta^4).$$

Answer: 125
user: Let $k \in Z_{>0}$ be the smallest positive integer with the property that $k\frac{gcd(x,y)gcd(y,z)}{lcm (x,y^2,z)}$ is a positive integer for all values $1 \le x \le y \le z \le 121$. If k' is the number of divisors of $k$, find the number of divisors of $k'$.

Answer: 174
user: An evil witch is making a potion to poison the people of PUMAClandia. In order for the potion to work, the number of poison dart frogs cannot exceed $5$, the number of wolves’ teeth must be an even number, and the number of dragon scales has to be a multiple of $6$. She can also put in any number of tiger nails. Given that the stew has exactly $2021$ ingredients, in how many ways can she add ingredients for her potion to work?

Answer: 1011
user: Have $b, c \in R$ satisfy $b \in (0, 1)$ and $c > 0$, then let $A,B$ denote the points of intersection of the line $y = bx+c$ with $y = |x|$, and let $O$ denote the origin of $R^2$. Let $f(b, c)$ denote the area of triangle $\vartriangle OAB$. Let $k_0 = \frac{1}{2022}$ , and for $n \ge 1$ let $k_n = k^2_{n-1}$. If the sum $\sum^{\infty}_{n=1}f(k_n, k_{n-1})$ can be written as $\frac{p}{q}$ for relatively prime positive integers $p, q$, find the remainder when $p+q$ is divided by 1000.

Answer: 484
user: Triangle $\vartriangle ABC$ has sidelengths $AB = 10$, $AC = 14$, and, $BC = 16$. Circle $\omega_1$ is tangent to rays $\overrightarrow{AB}$, $\overrightarrow{AC}$ and passes through $B$. Circle $\omega_2$ is tangent to rays $\overrightarrow{AB}$, $\overrightarrow{AC}$ and passes through $C$. Let $\omega_1$, $\omega_2$ intersect at points $X, Y$ . The square of the perimeter of triangle $\vartriangle AXY$ is equal to $\frac{a+b\sqrt{c}}{d}$ , where $a, b, c$, and, $d$ are positive integers such that $a$ and $d$ are relatively prime, and $c$ is not divisible by the square of any prime. Find $a + b + c + d$.

Answer: 6272
user: Pick $x, y, z$ to be real numbers satisfying $(-x+y+z)^2-\frac13 = 4(y-z)^2$, $(x-y+z)^2-\frac14 = 4(z-x)2$, and $(x+y-z)^2 -\frac15 = 4(x-y)^2$. If the value of $xy+yz +zx$ can be written as $\frac{p}{q}$ for relatively prime positive integers $p, q$, find $p + q$.

Answer: 1727
user: Let $\vartriangle ABC$ be an equilateral triangle. Points $D,E, F$ are drawn on sides $AB$,$BC$, and $CA$ respectively such that $[ADF] = [BED] + [CEF]$ and $\vartriangle ADF \sim \vartriangle BED \sim \vartriangle CEF$. The ratio $\frac{[ABC]}{[DEF]}$ can be expressed as $\frac{a+b\sqrt{c}}{d}$ , where $a$, $b$, $c$, and $d$ are positive integers such that $a$ and $d$ are relatively prime, and $c$ is not divisible by the square of any prime. Find $a + b + c + d$.
(Here $[P]$ denotes the area of polygon $P$.)

Answer: 17
user: Circle $\Gamma$ is centered at $(0, 0)$ in the plane with radius $2022\sqrt3$. Circle $\Omega$ is centered on the $x$-axis, passes through the point $A = (6066, 0)$, and intersects $\Gamma$ orthogonally at the point $P = (x, y)$ with $y > 0$. If the length of the minor arc $AP$ on $\Omega$ can be expressed as $\frac{m\pi}{n}$ forrelatively prime positive integers $m, n$, find $m + n$.
(Two circles intersect orthogonally at a point $P$ if the tangent lines at $P$ form a right angle.)

Answer: 1349
user: Patty is standing on a line of planks playing a game. Define a block to be a sequence of adjacent planks, such that both ends are not adjacent to any planks. Every minute, a plank chosen uniformly at random from the block that Patty is standing on disappears, and if Patty is standing on the plank, the game is over. Otherwise, Patty moves to a plank chosen uniformly at random within the block she is in; note that she could end up at the same plank from which she started. If the line of planks begins with $n$ planks, then for sufficiently large n, the expected number of minutes Patty lasts until the game ends (where the first plank disappears a minute after the game starts) can be written as $P(1/n)f(n) + Q(1/n)$, where $P,Q$ are polynomials and $f(n) =\sum^n_{i=1}\frac{1}{i}$ . Find $P(2023) + Q(2023)$.

Answer: 4045
user: A triangle $\vartriangle ABC$ is situated on the plane and a point $E$ is given on segment $AC$. Let $D$ be a point in the plane such that lines $AD$ and $BE$ are parallel. Suppose that $\angle EBC = 25^o$, $\angle BCA = 32^o$, and $\angle CAB = 60^o$. Find the smallest possible value of $\angle DAB$ in degrees.

Answer: 63
user: Let $\vartriangle ABC$ be a triangle with $AB = 5$, $BC = 8$, and, $CA = 7$. Let the center of the $A$-excircle be $O$, and let the $A$-excircle touch lines $BC$, $CA$, and,$ AB$ at points $X, Y$ , and, $Z$, respectively. Let $h_1$, $h_2$, and, $h_3$ denote the distances from $O$ to lines $XY$ , $Y Z$, and, ZX, respectively. If $h^2_1+ h^2_2+ h^2_3$ can be written as $\frac{m}{n}$ for relatively prime positive integers $m, n$, find $m + n$.

Answer: 2189
user: You’re given the complex number $\omega = e^{2i\pi/13} + e^{10i\pi/13} + e^{16i\pi/13} + e^{24i\pi/13}$, and told it’s a root of a unique monic cubic $x^3 +ax^2 +bx+c$, where $a, b, c$ are integers. Determine the value of $a^2 + b^2 + c^2$.

Answer: 18
user: Let $\alpha, \beta, \gamma \in C$ be the roots of the polynomial $x^3 - 3x2 + 3x + 7$. For any complex number $z$, let $f(z)$ be defined as follows:
$$f(z) = |z -\alpha | + |z - \beta|+ |z-\gamma | - 2 \underbrace{\max}_{w \in \{\alpha, \beta, \gamma}\} |z - w|.$$
Let $A$ be the area of the region bounded by the locus of all $z \in C$ at which $f(z)$ attains its global minimum. Find $\lfloor A \rfloor$.

Answer: 12
user: In the complex plane, let $z_1, z_2, z_3$ be the roots of the polynomial $p(x) = x^3- ax^2 + bx - ab$. Find the number of integers $n$ between $1$ and $500$ inclusive that are expressible as $z^4_1 +z^4_2 +z^4_3$ for some choice of positive integers $a, b$.

Answer: 51
user: Of all functions $h : Z_{>0} \to Z_{\ge 0}$, choose one satisfying $h(ab) = ah(b) + bh(a)$ for all $a, b \in Z_{>0}$ and $h(p) = p$ for all prime numbers $p$. Find the sum of all positive integers $n\le 100$ such that $h(n) = 4n$.

Answer: 729
user: A sequence of integers $x_1, x_2, ...$ is [i]double-dipped[/i] if $x_{n+2} = ax_{n+1} + bx_n$ for all $n \ge 1$ and some fixed integers $a, b$. Ri begins to form a sequence by randomly picking three integers from the set $\{1, 2, ..., 12\}$, with replacement. It is known that if Ri adds a term by picking anotherelement at random from $\{1, 2, ..., 12\}$, there is at least a $\frac13$ chance that his resulting four-term sequence forms the beginning of a double-dipped sequence. Given this, how many distinct three-term sequences could Ri have picked to begin with?

Answer: 84
user: Ryan Alweiss storms into the Fine Hall common room with a gigantic eraser and erases all integers $n$ in the interval $[2, 728]$ such that $3^t$ doesn’t divide $n!$, where $t = \left\lceil \frac{n-3}{2} \right\rceil$. Find the sum of the leftover integers in that interval modulo $1000$.

Answer: 11
user: Subsets $S$ of the first 3$5$ positive integers $\{1, 2, 3, ..., 35\}$ are called [i]contrived [/i] if $S$ has size $4$ and the sum of the squares of the elements of $S$ is divisible by $7$. Find the number of contrived sets.

Answer: 8605
user: Let $\vartriangle ABC$ be a triangle. Let $Q$ be a point in the interior of $\vartriangle ABC$, and let $X, Y,Z$ denote the feet of the altitudes from $Q$ to sides $BC$, $CA$, $AB$, respectively. Suppose that $BC = 15$, $\angle ABC = 60^o$, $BZ = 8$, $ZQ = 6$, and $\angle QCA = 30^o$. Let line $QX$ intersect the circumcircle of $\vartriangle XY Z$ at the point $W\ne X$. If the ratio $\frac{ WY}{WZ}$ can be expressed as $\frac{p}{q}$ for relatively prime positive integers $p, q$, find $p + q$.

Answer: 11
user: A triangle $\vartriangle A_0A_1A_2$ in the plane has sidelengths $A_0A_1 = 7$,$A_1A_2 = 8$,$A_2A_0 = 9$. For $i \ge 0$, given $\vartriangle A_iA_{i+1}A_{i+2}$, let $A_{i+3}$ be the midpoint of $A_iA_{i+1}$ and let Gi be the centroid of $\vartriangle A_iA_{i+1}A_{i+2}$. Let point $G$ be the limit of the sequence of points $\{G_i\}^{\infty}_{i=0}$. If the distance between $G$ and $G_0$ can be written as $\frac{a\sqrt{b}}{c}$ , where $a, b, c$ are positive integers such that $a$ and $c$ are relatively prime and $b$ is not divisible by the square of any prime, find $a^2 + b^2 + c^2$.

Answer: 422
user: For the function $$ g(a) = \underbrace{\max}_{x\in R} \left\{ \cos x + \cos \left(x + \frac{\pi}{6} \right)+ \cos \left(x + \frac{\pi}{4} \right) + cos(x + a) \right\},$$ let $b \in R$ be the input that maximizes $g$. If $\cos^2 b = \frac{m+\sqrt{n}+\sqrt{p}-\sqrt{q}}{24}$ for positive integers $m, n, p, q$, find $m + n + p + q$.

Answer: 54
user: Provided that $\{a_i\}^{28}_{i=1}$ are the $28$ distinct roots of $29x^{28} + 28x^{27} + ... + 2x + 1 = 0$, then the absolute value of $\sum^{28}_{i=1}\frac{1}{(1-a_i)^2}$ can be written as $\frac{p}{q}$ for relatively prime positive integers $p, q$. Find $p + q$.

Answer: 275
user: Observe the set $S =\{(x, y) \in Z^2 : |x| \le 5$ and $-10 \le y\le 0\}$. Find the number of points $P$ in $S$ such that there exists a tangent line from $P$ to the parabola $y = x^2 + 1$ that can be written in the form $y = mx + b$, where $m$ and $b$ are integers.

Answer: 15
user: Let $\vartriangle ABC$ be a triangle with $BC = 7$, $CA = 6$, and, $AB = 5$. Let $I$ be the incenter of $\vartriangle ABC$. Let the incircle of $\vartriangle ABC$ touch sides $BC$, $CA$, and $AB$ at points $D,E$, and $F$. Let the circumcircle of $\vartriangle AEF$ meet the circumcircle of $\vartriangle ABC$ for a second time at point $X\ne A$. Let $P$ denote the intersection of $XI$ and $EF$. If the product $XP \cdot IP$ can be written as $\frac{m}{n}$ for relatively prime positive integers $m, n$, find $m + n$.

Answer: 629
user: An ellipse has foci $A$ and $B$ and has the property that there is some point $C$ on the ellipse such that the area of the circle passing through $A$, $B$, and, $C$ is equal to the area of the ellipse. Let $e$ be the largest possible eccentricity of the ellipse. One may write $e^2$ as $\frac{a+\sqrt{b}}{c}$ , where $a, b$, and $c$ are integers such that $a$ and $c$ are relatively prime, and b is not divisible by the square of any prime. Find $a^2 + b^2 + c^2$.

Answer: 30
user: Let $\vartriangle ABC$ have sidelengths $BC = 7$, $CA = 8$, and, $AB = 9$, and let $\Omega$ denote the circumcircle of $\vartriangle ABC$. Let circles $\omega_A$, $\omega_B$, $\omega_C$ be internally tangent to the minor arcs $BC$, $CA$, $AB$ of $\Omega$, respectively, and tangent to the segments $BC$, $CA$, $AB$ at points $X$, $Y$, and, $Z$, respectively. Suppose that $\frac{BX}{XC} = \frac{CY}{Y A} = \frac{AZ}{ZB} = \frac12$ . Let $t_{AB}$ be the length of the common external tangent of $\omega_A$ and $\omega_B$, let $t_{BC}$ be the length of the common external tangent of $\omega_B$ and $\omega_C$, and let $t_{CA}$ be the length of the common external tangent of $\omega_C$ and $\omega_A$. If $t_{AB} + t_{BC} + t_{CA}$ can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m, n$, find $m + n$.

Answer: 59
user: A sequence of integers $a_1, a_2, \ldots, a_n$ is said to be [i]sub-Fibonacci[/i] if $a_1=a_2=1$ and $a_i \le a_{i-1}+a_{i-2}$ for all $3 \le i \le n.$ How many sub-Fibonacci sequences are there with $10$ terms such that the last two terms are both $20$?

Answer: 238
user: Let $\vartriangle ABC$ be a triangle with $\angle BAC = 90^o$, $\angle ABC = 60^o$, and $\angle BCA = 30^o$ and $BC = 4$. Let the incircle of $\vartriangle ABC$ meet sides $BC$, $CA$, $AB$ at points $A_0$, $B_0$, $C_0$, respectively. Let $\omega_A$, $\omega_B$, $\omega_C$ denote the circumcircles of triangles $\vartriangle B_0IC_0$ , $\vartriangle C_0IA_0$ , $\vartriangle A_0IB_0$, respectively. We construct triangle $T_A$ as follows: let $A_0B_0$ meet $\omega_B$ for the second time at $A_1\ne A_0$, let $A_0C_0$ meet $\omega_C$ for the second time at $A_2\ne A_0$, and let $T_A$ denote the triangle $\vartriangle A_0A_1A_2$. Construct triangles $T_B$, $T_C$ similarly. If the sum of the areas of triangles $T_A$, $T_B$, $T_C$ equals $\sqrt{m} - n$ for positive integers $m$, $n$, find $m + n$.

Answer: 15
user: Let $\vartriangle ABC$ be a triangle with $AB = 4$, $BC = 6$, and $CA = 5$. Let the angle bisector of $\angle BAC$ intersect $BC$ at the point $D$ and the circumcircle of $\vartriangle ABC$ again at the point $M\ne A$. The perpendicular bisector of segment $DM$ intersects the circle centered at $M$ passing through $B$ at two points, $X$ and $Y$ . Compute $AX \cdot AY$.

Answer: 36
user: The set of real values $a$ such that the equation $x^4-3ax^3+(2a^2+4a)x^2-5a^2x+3a^2$ has exactly two nonreal solutions is the set of real numbers between $x$ and $y,$ where $x<y.$ If $x+y$ can be written as $\tfrac{m}{n}$ for relatively prime positive integers $m,n,$ find $m+n.$

Answer: 8
user: Let $\triangle T B D$ be a triangle with $T B=6, B D=8$, and $D T=7$. Let $I$ be the incenter of $\triangle T B D$, and let $T I$ intersect the circumcircle of $\triangle T B D$ at $M \neq T$. Let lines $T B$ and $M D$ intersect at $Y$, and let lines $T D$ and $M B$ intersect at $X$. Let the circumcircles of $\triangle Y B M$ and $\triangle X D M$ intersect at $Z \neq M$. If the area of $\triangle Y B Z$ is $x$ and the area of $\triangle X D Z$ is $y$, then the ratio $\frac{x}{y}$ can be expressed as $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.

Answer: 97
user: Let $f(p)$ denote the number of ordered tuples $\left(x_{1}, x_{2}, \ldots, x_{p}\right)$ of nonnegative integers satisfying $\sum_{i=1}^{p} x_{i}=2022$, where $x_{i} \equiv i(\bmod p)$ for all $1 \leq i \leq p$. Find the remainder when $\sum_{p \in \mathcal{S}} f(p)$ is divided by 1000, where $\mathcal{S}$ denotes the set of all primes less than 2022 .

Answer: 475
user: Let $\vartriangle ABC$ be a triangle with $AB = 13$, $BC = 14$, and $CA = 15$. Let $D$, $E$, and $F$ be the midpoints of $AB$, $BC$, and $CA$ respectively. Imagine cutting $\vartriangle ABC$ out of paper and then folding $\vartriangle AFD$ up along $FD$, folding $\vartriangle BED$ up along $DE$, and folding $\vartriangle CEF$ up along $EF$ until $A$, $B$, and $C$ coincide at a point $G$. The volume of the tetrahedron formed by vertices $D$, $E$, $F$, and $G$ can be expressed as $\frac{p\sqrt{q}}{r}$ , where $p$, $q$, and $r$ are positive integers, $p$ and $r$ are relatively prime, and $q$ is square-free. Find $p + q + r$.

Answer: 80
user: Amir enters Fine Hall and sees the number $2$ written on the blackboard. Amir can perform the following operation: he flips a coin, and if it is heads, he replaces the number $x$ on the blackboard with $3x+1;$ otherwise, he replaces $x$ with $\lfloor x/3\rfloor.$ If Amir performs the operation four times, let $\tfrac{m}{n}$ denote the expected number of times that he writes the digit $1$ on the blackboard, where $m,n$ are relatively prime positive integers. Find $m+n.$

Answer: 27
user: Let $P(x)$ be a polynomial with integer coefficients satisfying $$(x^2+1)P(x-1)=(x^2-10x+26)P(x)$$ for all real numbers $x.$ Find the sum of all possible values of $P(0)$ between $1$ and $5000,$ inclusive.

Answer: 5100
user: Let $S$ be the set of degree $4$ polynomials $f$ with complex number coefficients satisfying $f(1)=f(2)^2=f(3)^3$ $=$ $f(4)^4=f(5)^5=1.$ Find the mean of the fifth powers of the constant terms of all the members of $S.$

Answer: 1643751
user: The sum $\sum_{k=1}^{2020} k \cos \left(\frac{4 k \pi}{4041}\right)$ can be written in the form

$$
\frac{a \cos \left(\frac{p \pi}{q}\right)-b}{c \sin ^{2}\left(\frac{p \pi}{q}\right)}
$$

where $a, b, c$ are relatively prime positive integers and $p, q$ are relatively prime positive integers where $p<q$. Determine $a+b+c+p+q$.

Answer: 4049
user: A spider is walking on the boundary of equilateral triangle $\triangle{ABC}$ (vertices labelled in counterclockwise order), starting at vertex $A$. Each second, she moves to one of her two adjacent vertices with equal probability. The windiness of a path that starts and ends at $A$ is the net number of counterclockwise revolutions made. For example, the windiness of the path $ABCA$ is $1,$ and the windiness of the path $ABCACBACBA$ is $-1$. What is the remainder modulo $1000$ of the sum of the squares of the windiness values taken over all possible paths that end back at vertex $A$ after $2025$ seconds?

Answer: 50
user: Let $\Gamma_{1}$ and $\Gamma_{2}$ be externally tangent circles with radii $\frac{1}{2}$ and $\frac{1}{8}$, respectively. The line $\ell$ is a common external tangent to $\Gamma_{1}$ and $\Gamma_{2}$. For $n \geq 3$, we define $\Gamma_{n}$ as the smallest circle tangent to $\Gamma_{n-1}, \Gamma_{n-2}$, and $\ell$. The radius of $\Gamma_{10}$ can be expressed as $\frac{a}{b}$ where $a, b$ are relatively prime positive integers. Find $a+b$.

Answer: 15843
user: Let $a_{n}$ denote the number of ternary strings of length $n$ so that there does not exist a $k<n$ such that the first $k$ digits of the string equals the last $k$ digits. What is the largest integer $m$ such that $3^{m} \mid a_{2023}$ ?

Answer: 9
user: For a positive integer $n,$ let $P_n$ be the set of sequences of $2n$ elements, each $0$ or $1,$ where there are exactly $n$ $1$’s and $n$ $0$’s. I choose a sequence uniformly at random from $P_n.$ Then, I partition this sequence into maximal blocks of consecutive $0$’s and $1$’s. Define $f(n)$ to be the expected value of the sum of squares of the block lengths of this uniformly random sequence. What is the largest integer value that $f(n)$ can take on?

Answer: 121
user: Let $\vartriangle ABC$ have $AB = 15$, $AC = 20$, and $BC = 21$. Suppose $\omega$ is a circle passing through $A$ that is tangent to segment $BC$. Let point $D\ne A$ be the second intersection of AB with $\omega$, and let point $E \ne A$ be the second intersection of $AC$ with $\omega$. Suppose $DE$ is parallel to $BC$. If $DE = \frac{a}{b}$ , where $a$, $b$ are relatively prime positive integers, find $a + b$.

Answer: 361
user: Let $\triangle{ABC}$ be an isosceles triangle with $AB = AC =\sqrt{7}, BC=1$. Let $G$ be the centroid of $\triangle{ABC}$. Given $ j\in \{0,1,2\}$, let $T_{j}$ denote the triangle obtained by rotating $\triangle{ABC}$ about $G$ by $\frac{2\pi j}{3}$ radians. Let $\mathcal{P}$ denote the intersection of the interiors of triangles $T_0,T_1,T_2$. If $K$ denotes the area of $\mathcal{P}$, then $K^2=\frac{a}{b}$ for relatively prime positive integers $a, b$. Find $a + b$.

Answer: 1843