ChenDRAG/VeRL_math_validation huggingface dataset:
Problem: A magician has a hat that contains $a$ white rabbits and $b$ black rabbits. The magician repeatedly draws pairs of rabbits chosen at random from the hat, without replacement. Call a pair of rabbits *checkered* if it consists of one white rabbit and one black rabbit. Given that the magician eventually draws out all the rabbits without ever drawing out an unpaired rabbit and that the expected value of the number of checkered pairs that the magician draws is $2020$, compute the number of possible pairs $(a,b)$.
Answer: 16
Problem: A sequence of real numbers $x_0, x_1, x_2, \ldots$ is defined as follows: $x_0 = 1989$ and for each $n \geq 1$,
\[ x_n = -\frac{1989}{n} \sum^{n-1}_{k=0} x_k. \]
Calculate the value of $\sum^{1989}_{n=0} 2^n x_n.$
Answer: -1989
Problem: Let $C$ and $D$ be points inside angle $\angle AOB$ such that $5\angle COD = 4\angle AOC$ and $3\angle COD = 2\angle DOB$. If $\angle AOB = 105^{\circ}$, find $\angle COD$.
Answer: 28
Problem: Find the number of $10$-digit palindromes that are not divisible by $11$.
Answer: 0
Problem: Plusses and minuses are inserted in the expression:
\[ \pm 1 \pm 2 \pm 3 \dots \pm 2016 \]
such that when evaluated, the result is divisible by 2017. Let there be $N$ ways for this to occur. Compute the remainder when $N$ is divided by 503.
Answer: 52
Problem: Find the number of integer solutions of $\left[\frac{x}{100} \left[\frac{x}{100}\right]\right]= 5$.
Answer: 50
Problem: In a rectangle $ABCD$, let $M$ and $N$ be the midpoints of sides $BC$ and $CD$, respectively, such that $AM$ is perpendicular to $MN$. Given that the length of $AN$ is $60$, the area of rectangle $ABCD$ is $m \sqrt{n}$ for positive integers $m$ and $n$ such that $n$ is not divisible by the square of any prime. Compute $100m+n$.
Answer: 160002
Problem: Call a polynomial $f$ with positive integer coefficients "triangle-compatible" if any three coefficients of $f$ satisfy the triangle inequality. For instance, $3x^3 + 4x^2 + 6x + 5$ is triangle-compatible, but $3x^3 + 3x^2 + 6x + 5$ is not. Given that $f$ is a degree $20$ triangle-compatible polynomial with $-20$ as a root, what is the least possible value of $f(1)$?
Answer: 420
Problem: For positive integers $a$ and $b$, the operation $a \uparrow\uparrow b$ is defined as follows:  
- $a \uparrow\uparrow 1 = a$  
- $a \uparrow\uparrow b = a^{a \uparrow\uparrow (b-1)}$ for $b > 1$.  

Find the smallest positive integer $n$ for which there exists a positive integer $a$ such that $a \uparrow\uparrow 6 \not\equiv a \uparrow\uparrow 7 \pmod{n}$.
Answer: 2
Problem: What is the smallest non-square positive integer that is the product of four prime numbers (not necessarily distinct)?
Answer: 24
Problem: There exists one pair of positive integers $a, b$ such that $100 > a > b > 0$ and $\frac{1}{a} + \frac{1}{b} = \frac{2}{35}$. Find $a + b$.
Answer: 72
Problem: Find the minimum natural number $n$ with the following property: between any collection of $n$ distinct natural numbers in the set $\{1,2, \dots,999\}$, it is possible to choose four different $a, b, c, d$ such that $a + 2b + 3c = d$.
Answer: 835
Problem: A positive integer $n$ between $1$ and $N = 2007^{2007}$ inclusive is selected at random. If $a$ and $b$ are natural numbers such that $\frac{a}{b}$ is the probability that $N$ and $n^3 - 36n$ are relatively prime, find the value of $a + b$.
Answer: 1109
Problem: Find the minimum value of $2x^2 + 2y^2 + 5z^2 - 2xy - 4yz - 4x - 2z + 15$ for real numbers $x$, $y$, $z$.
Answer: 10
Problem: A $10$-digit number is called interesting if its digits are distinct and it is divisible by $11111$. Find the number of interesting numbers.
Answer: 3456
Problem: Carrie, Miranda, Charlotte, and Samantha are sitting at a table with $5$ numbered chairs (numbered $1$ through $5$). One chair is left open for Big, should he decide to join the four for lunch. In how many distinct ways can the four women occupy the table?
Answer: 120
Problem: The dollar is now worth $\frac{1}{980}$ ounce of gold. After the $n^{th}$ 7001 billion dollars bailout package passed by congress, the dollar gains $\frac{1}{2^{2^{n-1}}}$ of its $(n-1)^{th}$ value in gold. After four bank bailouts, the dollar is worth $\frac{1}{b}(1-\frac{1}{2^c})$ in gold, where $b, c$ are positive integers. Find $b + c$.
Answer: 506
Problem: For a regular hexahedron and a regular octahedron, all their faces are regular triangles with each side of length $a$. Let their inradii be $r_1$ and $r_2$, respectively. If $\frac{r_1}{r_2} = \frac{m}{n}$, where $\gcd(m, n) = 1$, then find the value of $mn$.
Answer: 2
Problem: The remainder when $x^{100} - x^{99} + \ldots - x + 1$ is divided by $x^2 - 1$ can be written in the form $ax + b$. Find $2a + b$.
Answer: -49
Problem: A triangle has sides of length $\sqrt{13}$, $\sqrt{17}$, and $2 \sqrt{5}$. Compute the area of the triangle.
Answer: 7
Problem: Jack is jumping on the number line. He first jumps one unit and every jump after that, he jumps one unit more than the previous jump. What is the least number of jumps it takes to reach exactly $19999$ from his starting place?
Answer: 201
Problem: Knowing that the system:
\[
\begin{align*}
x + y + z &= 3, \\
x^3 + y^3 + z^3 &= 15, \\
x^4 + y^4 + z^4 &= 35,
\end{align*}
\]
has a real solution \(x, y, z\) for which \(x^2 + y^2 + z^2 < 10\), find the value of \(x^5 + y^5 + z^5\) for that solution.
Answer: 83
Problem: A rectangle has an area of $16$ and a perimeter of $18$. Determine the length of the diagonal of the rectangle.
Answer: 7
Problem: Find the number of non-congruent scalene triangles whose sides all have integral lengths, and the longest side has length $11$.
Answer: 20
Problem: From the positive integers, $m, m+1, \dots, m+n$, only the sum of digits of $m$ and the sum of digits of $m+n$ are divisible by $8$. Find the maximum value of $n$.
Answer: 15
Problem: Let $\mathcal{S}$ be a regular 18-gon. For two vertices in $\mathcal{S}$, define the \textit{distance} between them as the length of the shortest path along the edges of $\mathcal{S}$ between them (e.g., adjacent vertices have distance 1). Find the number of ways to choose three distinct vertices from $\mathcal{S}$ such that no two of them have distances 1, 8, or 9.
Answer: 240
Problem: A triangle $\triangle A_0A_1A_2$ in the plane has side lengths $A_0A_1 = 7$, $A_1A_2 = 8$, $A_2A_0 = 9$. For $i \ge 0$, given $\triangle A_iA_{i+1}A_{i+2}$, let $A_{i+3}$ be the midpoint of $A_iA_{i+1}$ and let $G_i$ be the centroid of $\triangle A_iA_{i+1}A_{i+2}$. 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
Problem: Find the sum of all distinct possible values of $x^2 - 4x + 100$, where $x$ is an integer between 1 and 100, inclusive.
Answer: 328053
Problem: There are $18$ children in the class. Parents decided to give the children a cake. To do this, they first learned from each child the area of the piece they want to get. After that, they showed a square-shaped cake, the area of which is exactly equal to the sum of the $18$ named numbers. However, when they saw the cake, the children wanted their pieces to be squares too. The parents cut the cake with lines parallel to the sides of the cake (cuts do not have to start or end on the side of the cake). For what maximum $k$ are the parents guaranteed to cut out $k$ square pieces from the cake, which can be given to $k$ children so that each of them gets what they want?
Answer: 12
Problem: A positive integer $m$ has the property that $m^2$ can be expressed in the form $4n^2 - 5n + 16$, where $n$ is an integer (of any sign). Find the maximum value of $|m - n|$.
Answer: 14
Problem: Find the number of fractions in the following list that are in their lowest terms (i.e., for $\frac{p}{q}$, $\gcd(p,q) = 1$):

$$\frac{1}{2014}, \frac{2}{2013}, \dots, \frac{1007}{1008}$$
Answer: 720
Problem: 7. Let $r_1, r_2, r_3$ be the roots of the polynomial equation $x^3 - 2x^2 + 4x + 10 = 0$. Compute the value of $(r_1 + 2)(r_2 + 2)(r_3 + 2)$.
Answer: 14
Problem: Let $A=\{1, 2, \ldots, 2006\}$. Find the maximal number of subsets of $A$ that can be chosen such that the intersection of any two distinct subsets has 2004 elements.
Answer: 2006
Problem: The entries of a $7 \times 5$ table are filled with numbers such that in each $2 \times 3$ rectangle (either vertical or horizontal), the sum of the numbers is $0$. For $100$ dollars, Peter may choose any single entry and learn the number in it. What is the least amount of dollars he should spend to determine the total sum of numbers in the table for sure?
Answer: 0
Problem: Let $a_1 = 24$ and form the sequence $a_n$, for $n \geq 2$, by the recurrence relation $a_n = 100a_{n-1} + 134$. The first few terms of the sequence are:
\[
24, 2534, 253534, 25353534, \ldots
\]
Determine the smallest value of $n$ for which $a_n$ is divisible by $99$.
Answer: 88
Problem: A quarry wants to sell a large pile of gravel. At full price, the gravel would sell for $3200$ dollars. During the first week, the quarry sells $60\%$ of the gravel at full price. The following week, the quarry reduces the price by $10\%$, and again, it sells $60\%$ of the remaining gravel. Each subsequent week, the quarry continues to reduce the price by another $10\%$ and sells $60\%$ of the remaining gravel. This process continues until only a handful of gravel is left. How many dollars does the quarry collect for the sale of all its gravel?
Answer: 3000
Problem: There are relatively prime positive integers $m$ and $n$ such that the parabola with equation $y = 4x^2$ is tangent to the parabola with equation $x = y^2 + \frac{m}{n}$. Find $m + n$.
Answer: 19
Problem: Evaluate the sum $\left\lfloor\log_2 1\right\rfloor + \left\lfloor\log_2 2\right\rfloor + \left\lfloor\log_2 3\right\rfloor + \ldots + \left\lfloor\log_2 256\right\rfloor$.
Answer: 1546
Problem: The following diagram shows an equilateral triangle and two squares that share common edges. The area of each square is $75$. Find the distance from point $A$ to point $B$.

```latex
\begin{asy}
size(175);
defaultpen(linewidth(0.8));
pair A=(-3,0), B=(3,0), C=rotate(60,A)*B, D=rotate(270,B)*C, E=rotate(90,C)*B, F=rotate(270,C)*A, G=rotate(90,A)*C;
draw(A--G--F--C--A--B--C--E--D--B);
label("$A$", F, N);
label("$B$", E, N);
\end{asy}
```
Answer: 15
Problem: Let $ABCD$ be a convex quadrilateral with $AB = AD$, $m\angle A = 40^{\circ}$, $m\angle C = 130^{\circ}$, and $m\angle ADC - m\angle ABC = 20^{\circ}$. Find the measure of the non-reflex angle $\angle CDB$ in degrees.
Answer: 35
Problem: Let $\triangle ABC$ have $AB = 14$, $BC = 30$, $AC = 40$ and $\triangle AB'C'$ with $AB' = 7\sqrt{6}$, $B'C' = 15\sqrt{6}$, $AC' = 20\sqrt{6}$ such that $\angle BAB' = \frac{5\pi}{12}$. The lines $BB'$ and $CC'$ intersect at point $D$. Let $O$ be the circumcenter of $\triangle BCD$, and let $O'$ be the circumcenter of $\triangle B'C'D$. Then the length of segment $OO'$ 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 $a+b+c+d$.
Answer: 55
Problem: Determine the largest positive integer that is a divisor of all three of \( A = 2^{2010} \times 3^{2010} \), \( B = 3^{2010} \times 5^{2010} \), and \( C = 5^{2010} \times 2^{2010} \).
Answer: 1
Problem: A function $f$ from the positive integers to the nonnegative integers is defined recursively by $f(1) = 0$ and $f(n+1) = 2^{f(n)}$ for every positive integer $n$. What is the smallest $n$ such that $f(n)$ exceeds the number of atoms in the observable universe (approximately $10^{80}$)?
Answer: 7
Problem: We define the weight $W$ of a positive integer as follows: 
1. $W(1) = 0$
2. $W(2) = 1$
3. $W(p) = 1 + W(p + 1)$ for every odd prime $p$
4. $W(c) = 1 + W(d)$ for every composite $c$, where $d$ is the greatest proper factor of $c$.

Compute the greatest possible weight of a positive integer less than 100.
Answer: 12
Problem: Let $a, b, c$ be the roots of the equation $x^3 - 9x^2 + 11x - 1 = 0$, and define $s = \sqrt{a} + \sqrt{b} + \sqrt{c}$. Compute $s^4 - 18s^2 - 8s$.
Answer: -37
Problem: 99 identical balls lie on a table. 50 balls are made of copper, and 49 balls are made of zinc. The assistant numbered the balls. Once a spectrometer test is applied to 2 balls, it allows determining whether they are made of the same metal or not. However, the results of the test can be obtained only the next day. What is the minimum number of tests required to determine the material of each ball if all the tests should be performed today?
Answer: 98
Problem: Geodude wants to assign one of the integers $1,2,3,\ldots,11$ to each lattice point $(x,y,z)$ in a 3D Cartesian coordinate system. In how many ways can Geodude do this if for every lattice parallelogram $ABCD$, the positive difference between the sum of the numbers assigned to $A$ and $C$ and the sum of the numbers assigned to $B$ and $D$ must be a multiple of $11$? 

A lattice point is a point with all integer coordinates. A lattice parallelogram is a parallelogram with all four vertices lying on lattice points. Here, we say four not necessarily distinct points $A,B,C,D$ form a parallelogram $ABCD$ if and only if the midpoint of segment $AC$ coincides with the midpoint of segment $BD$. 

Clarifications:
- The "positive difference" between two real numbers $x$ and $y$ is the quantity $|x-y|$. Note that this may be zero.
- The last sentence was added to remove confusion about "degenerate parallelograms."
Answer: 14641
Problem: Emily Thorne is hosting a Memorial Day Party to kick off the summer in the Hamptons. She needs to figure out the seating arrangement for her guests. Here are the conditions:

- If she seats $4$ guests per table, there is $1$ guest left over.
- If she seats $5$ guests per table, there are $3$ guests left over.
- If she seats $6$ guests per table, there is $1$ guest left over.

Given that the number of guests is at least $100$ but no more than $200$, what is the greatest possible number of guests?
Answer: 193
Problem: A $10$-digit number is called interesting if its digits are distinct and the number is divisible by $11111$. Determine the total number of interesting numbers.
Answer: 3456
Problem: Let $ABCD$ be a square with side length $16$ and center $O$. Let $\mathcal{S}$ be the semicircle with diameter $AB$ that lies outside of $ABCD$, and let $P$ be a point on $\mathcal{S}$ such that $OP = 12$. Compute the area of triangle $CDP$.
Answer: 136
Problem: Two concentric circles have radii $1$ and $4$. Six congruent circles form a ring where each of the six circles is tangent to the two circles adjacent to it as shown. The three lightly shaded circles are internally tangent to the circle with radius $4$ while the three darkly shaded circles are externally tangent to the circle with radius $1$. The radius of the six congruent circles can be written $\frac{k+\sqrt{m}}{n}$, where $k, m,$ and $n$ are integers with $k$ and $n$ relatively prime. Find $k+m+n$.
Answer: 126
Problem: Let $\mathbb{Z}^{\geq 0}$ be the set of all non-negative integers. Consider a function $f: \mathbb{Z}^{\geq 0} \to \mathbb{Z}^{\geq 0}$ such that $f(0) = 1$ and $f(1) = 1$, and for any integer $n \geq 1$, we have:
\[
f(n + 1)f(n - 1) = nf(n)f(n - 1) + (f(n))^2.
\]
Determine the value of $\frac{f(2023)}{f(2022)}$.
Answer: 2045254
Problem: Find the sum of all positive integers less than $2006$ which satisfy the following conditions:
1. The number is a multiple of six.
2. The number is one more than a multiple of seven.
Answer: 47094
Problem: Let $x, y, z, w$ be integers such that $2^x + 2^y + 2^z + 2^w = 24.375$. Find the value of $xyzw$.
Answer: 72
Problem: Find the largest prime $p$ such that there exist positive integers $a,b$ satisfying $$p=\frac{b}{2}\sqrt{\frac{a-b}{a+b}}.$$
Answer: 5
Problem: The odometer of a family car shows \(15,951\) miles. The driver noticed that this number is palindromic: it reads the same backward as forwards. "Curious," the driver said to himself, "it will be a long time before that happens again." Surprised, he saw his third palindromic odometer reading (not counting \(15,951\)) exactly five hours later. How many miles per hour was the car traveling in those 5 hours (assuming speed was constant)?
Answer: 62
Problem: Determine the number of integers $2 \le n \le 2016$ such that $n^n - 1$ is divisible by $2$, $3$, $5$, and $7$.
Answer: 9
Problem: Determine the number of sequences of positive integers $1 = x_0 < x_1 < \dots < x_{10} = 10^{5}$ with the property that for each $m=0,\dots,9$, the number $\frac{x_{m+1}}{x_m}$ is a prime number.
Answer: 252
Problem: Pratyya and Payel each have a number, $n$ and $m$ respectively, where $n > m$. Every day, Pratyya multiplies his number by $2$ and then subtracts $2$ from it, while Payel multiplies his number by $2$ and then adds $2$ to it. In other words, on the first day, their numbers will be $(2n - 2)$ and $(2m + 2)$ respectively. Find the minimum integer $x$ with proof such that if $n - m \geq x$, then Pratyya's number will be larger than Payel's number every day.
Answer: 4
Problem: Suppose $a_{1} < a_{2}< \cdots < a_{2024}$ is an arithmetic sequence of positive integers, and $b_{1} <b_{2} < \cdots <b_{2024}$ is a geometric sequence of positive integers. Find the maximum possible number of integers that could appear in both sequences, over all possible choices of the two sequences.
Answer: 11
Problem: We have four registers, $R_1, R_2, R_3, R_4$, such that $R_i$ initially contains the number $i$ for $1 \leq i \leq 4$. We are allowed two operations:

1. Simultaneously swap the contents of $R_1$ and $R_3$ as well as $R_2$ and $R_4$.

2. Simultaneously transfer the contents of $R_2$ to $R_3$, the contents of $R_3$ to $R_4$, and the contents of $R_4$ to $R_2$. For example, if we do this once, then $(R_1, R_2, R_3, R_4) = (1, 4, 2, 3)$.

Using these two operations as many times as desired and in whatever order, what is the total number of possible outcomes?
Answer: 12
Problem: If $ABCDE$ is a regular pentagon and $X$ is a point in its interior such that $\triangle CDX$ is equilateral, compute $\angle AXE$ in degrees.
Answer: 84
Problem: Find the number of all $6$-digit natural numbers such that the sum of their digits is $10$ and each of the digits $0, 1, 2, 3$ occurs at least once in them.
Answer: 490
Problem: In a circus, there are $n$ clowns who dress and paint themselves using a selection of 12 distinct colours. Each clown is required to use at least five different colours. One day, the ringmaster of the circus orders that no two clowns have exactly the same set of colours and no more than 20 clowns may use any one particular colour. Find the largest number $n$ of clowns so as to make the ringmaster's order possible.
Answer: 48
Problem: How many four-digit positive integers have exactly one digit equal to $1$ and exactly one digit equal to $3$?
Answer: 720
Problem: In a quadrilateral $ABCD$, it is given that $AB = AD = 13$, $BC = CD = 20$, and $BD = 24$. If $r$ is the radius of the circle inscribable in the quadrilateral, what is the integer closest to $r$?
Answer: 8
Problem: Esmeralda has created a special knight to play on quadrilateral boards that are identical to chessboards. If a knight is in a square, it can move to another square by moving 1 square in one direction and 3 squares in a perpendicular direction (which is a diagonal of a $2 \times 4$ rectangle instead of $2 \times 3$ like in chess). In this movement, it doesn't land on the squares between the beginning square and the final square it lands on.

A trip of the length $n$ of the knight is a sequence of $n$ squares $C_1, C_2, \ldots, C_n$ which are all distinct such that the knight starts at the $C_1$ square and for each $i$ from $1$ to $n-1$, it can use the movement described before to go from the $C_i$ square to the $C_{i+1}$.

Determine the greatest $N \in \mathbb{N}$ such that there exists a path of the knight with length $N$ on a $5 \times 5$ board.
Answer: 12
Problem: Naomi has three colors of paint which she uses to paint the pattern below. She paints each region a solid color, and each of the three colors is used at least once. If Naomi is willing to paint two adjacent regions with the same color, how many color patterns could Naomi paint?

```asy
size(150);
defaultpen(linewidth(2));
draw(origin--(37,0)--(37,26)--(0,26)--cycle^^(12,0)--(12,26)^^(0,17)--(37,17)^^(20,0)--(20,17)^^(20,11)--(37,11));
```
Answer: 540
Problem: Let \( \triangle ABC \) be a triangle with \( \angle A = 90^\circ \). The bisectors of angles \( B \) and \( C \) intersect at point \( I \). Suppose that \( IH \) is perpendicular to \( BC \) and \( H \) lies on \( BC \). If \( HB = 5 \text{ cm} \) and \( HC = 8 \text{ cm} \), compute the area of \( \triangle ABC \).
Answer: 40
Problem: Lucía multiplies some positive one-digit numbers (not necessarily distinct) and obtains a number $n$ greater than 10. Then, she multiplies all the digits of $n$ and obtains an odd number. Find all possible values of the units digit of $n$.
Answer: 5
Problem: Find the number of even integers \( n \) such that \( 0 \le n \le 100 \) and \( 5 \mid n^2 \cdot 2^{2n^2} + 1 \).
Answer: 20
Problem: Find the number of ordered quadruples of positive integers \((a,b,c,d)\) such that \(ab + cd = 10\).
Answer: 58
Problem: Determine the last four decimal digits of the number $1997 \cdot 5^{1998}$.
Answer: 3125
Problem: Find the highest power of $2$ that divides exactly into $1996! = 1 \times 2 \times \cdots \times 1996$.
Answer: 1989
Problem: The set $A=\{1,2,3,\cdots, 10\}$ contains the numbers $1$ through $10$. A subset of $A$ of size $n$ is competent if it contains $n$ as an element. A subset of $A$ is minimally competent if it itself is competent, but none of its proper subsets are. Find the total number of minimally competent subsets of $A$.
Answer: 55
Problem: Let there be 50 natural numbers $a_i$ such that $0 < a_1 < a_2 < \ldots < a_{50} < 150$. What is the greatest possible sum of the differences $d_j$ where each $d_j = a_{j+1} - a_j$?
Answer: 148
Problem: Define a finite sequence $\left( s_i \right)_{1\le i\le 2004}$ with initial conditions $s_0 + 2 = s_1 + 1 = s_2 = 2$ and the recurrence relation:
$$ s_n = 1 + s_{n-1} + s_{n-2} - s_{n-3}. $$
Calculate its last element.
Answer: 1005006
Problem: Compute the largest positive integer $n$ such that $\frac{2007!}{2007^n}$ is an integer.
Answer: 9
Problem: For $n$ a positive integer, denote by $P(n)$ the product of all positive integer divisors of $n$. Find the smallest $n$ for which 

\[ P(P(P(n))) > 10^{12} \]
Answer: 6
Problem: A contest consists of six problems, each worth 7 points. For any given problem, a contestant can score either $0$, $1$, or $7$ points. Determine the number of possible total scores a contestant can achieve across all six problems.
Answer: 28
Problem: Let $P$ and $Q$ be points on $AC$ and $AB$, respectively, of triangle $\triangle ABC$ such that $PB=PC$ and $PQ \perp AB$. Suppose $\frac{AQ}{QB}=\frac{AP}{PB}$. Find $\angle CBA$, in degrees.
Answer: 90
Problem: Let $a_1 < a_2 < \cdots < a_k$ denote the sequence of all positive integers between $1$ and $91$ which are relatively prime to $91$, and set $\omega = e^{2\pi i/91}$. Define \[S = \prod_{1\leq q < p\leq k}(\omega^{a_p} - \omega^{a_q}).\] Given that $S$ is a positive integer, compute the number of positive divisors of $S$.
Answer: 1054
Problem: The side lengths of a triangle are natural number multiples of $7$, and each side is smaller than $40$. How many triangles satisfy these conditions?
Answer: 22
Problem: For an olympiad geometry problem, Tina wants to draw an acute triangle whose angles each measure a multiple of $10^\circ$. She doesn't want her triangle to have any special properties, so none of the angles can measure $30^\circ$ or $60^\circ$, and the triangle should definitely not be isosceles.

How many different triangles can Tina draw? (Similar triangles are considered the same.)
Answer: 0