ChenDRAG/VeRL_math_validation huggingface dataset:
Problem: In a drawer, there are at most $2009$ balls. Some of these balls are white, and the rest are blue, randomly distributed. If two balls are taken at the same time, the probability that both balls are either blue or both white is $\frac{1}{2}$. Determine the maximum number of white balls in the drawer such that the probability statement holds true.
Answer: 990
Problem: $f(x)$ is a quartic polynomial with a leading coefficient $1$ where $f(2)=4$, $f(3)=9$, $f(4)=16$, and $f(5)=25$. Compute $f(8)$.
Answer: 424
Problem: In a country, there are $28$ cities, and between some cities, there are two-way flights. Each city has exactly one airport, which can be either small, medium, or big. For every route that:
1. Contains more than two cities,
2. Does not repeat any city,
3. Ends where it begins,

It must include all types of airports. What is the maximum number of flights in this country?
Answer: 53
Problem: Alex the Kat has written $61$ problems for a math contest, and there are a total of $187$ problems submitted. How many more problems does he need to write and submit before he has written half of the total problems?
Answer: 65
Problem: Find the greatest possible sum of integers $a$ and $b$ such that $\frac{2021!}{20^a \cdot 21^b}$ is a positive integer.
Answer: 837
Problem: Compute the number of ordered triples \((x, y, z)\) of integers satisfying 
\[x^2 + y^2 + z^2 = 9.\]
Answer: 30
Problem: Evaluate $\lim_{n \to \infty} n^2 \int_{-\frac{1}{n}}^{\frac{1}{n}} (2005\sin x + 2006\cos x)|x| \, dx.$
Answer: 2006
Problem: How many non-congruent isosceles triangles, including equilateral triangles, have positive integer side lengths and a perimeter less than 20?
Answer: 41
Problem: Show that there exists an infinite arithmetic progression of natural numbers such that the first term is $16$ and the number of positive divisors of each term is divisible by $5$. Of all such sequences, find the one with the smallest possible common difference.
Answer: 32
Problem: In triangle $ABC$, $AB = 52$, $BC = 34$, and $CA = 50$. The side $BC$ is divided into $n$ equal segments by placing $n-1$ new points. Among these points are the feet of the altitude, median, and angle bisector from vertex $A$. What is the smallest possible value of $n$?
Answer: 102
Problem: Find the numerator of \(\frac{1010\overbrace{11 \ldots 11}^{2011 \text{ ones}}0101}{1100\underbrace{11 \ldots 11}_{2011\text{ ones}}0011}\) when reduced.
Answer: 9091
Problem: Let $ABCD$ be an isosceles trapezoid with $AD \parallel BC$ and $BC > AD$ such that the distance between the incenters of $\triangle ABC$ and $\triangle DBC$ is $16$. If the perimeters of $ABCD$ and $ABC$ are $120$ and $114$ respectively, then the area of $ABCD$ can be written as $m\sqrt{n}$, where $m$ and $n$ are positive integers with $n$ not divisible by the square of any prime. Find $100m+n$.
Answer: 7114
Problem: Anders is solving a math problem, and he encounters the expression $\sqrt{15!}$. He attempts to simplify this radical as $a\sqrt{b}$ where $a$ and $b$ are positive integers. The sum of all possible values of $ab$ can be expressed in the form $q \cdot 15!$ for some rational number $q$. Find $q$.
Answer: 4
Problem: Two quadratic polynomials $A(x)$ and $B(x)$ have a leading term of $x^2$. For some real numbers $a$ and $b$, the roots of $A(x)$ are $1$ and $a$, and the roots of $B(x)$ are $6$ and $b$. If the roots of $A(x) + B(x)$ are $a + 3$ and $b + \frac{1}{2}$, then compute $a^2 + b^2$.
Answer: 3
Problem: Ted loves creating haikus (Japanese three-line poems with $5$, $7$, $5$ syllables each), but his vocabulary is rather limited. For integers $1 \le n \le 7$, he knows $2n$ words with $n$ syllables. Words cannot cross between lines but may be repeated. Compute the number of distinct haikus Ted can make, and find the remainder when this number is divided by $100$.
Answer: 28
Problem: Which is the smallest natural number $n$ such that $n!$ has at least 2013 divisors?
Answer: 14
Problem: Amandine and Brennon play a turn-based game, with Amandine starting. On their turn, a player must select a positive integer which cannot be represented as a sum of multiples of any of the previously selected numbers. For example, if $3, 5$ have been selected so far, only $1, 2, 4, 7$ are available to be picked; if only $3$ has been selected so far, all numbers not divisible by three are eligible. A player loses immediately if they select the integer $1$. 

Call a number $n$ \textit{feminist} if $\gcd(n, 6) = 1$ and if Amandine wins if she starts with $n$. Compute the sum of the \textit{feminist} numbers less than $40$.
Answer: 192
Problem: There is a sequence of numbers $+1$ and $-1$ of length $n$. It is known that the sum of every $10$ neighboring numbers in the sequence is $0$, and that the sum of every $12$ neighboring numbers in the sequence is not zero. What is the maximal value of $n$?
Answer: 15
Problem: A positive integer is written on each corner of a square such that numbers on opposite vertices are relatively prime, while numbers on adjacent vertices are not relatively prime. What is the smallest possible value of the sum of these $4$ numbers?
Answer: 60
Problem: Let $ABCD$ be a square with $AB=6$. A point $P$ in the interior is $2$ units away from side $BC$ and $3$ units away from side $CD$. What is the distance from $P$ to $A$?
Answer: 5
Problem: Let $\alpha ,\ \beta$ be the distinct positive roots of the equation $2x=\tan x$. Evaluate the following definite integral: 
\[ \int_{0}^{1}\sin(\alpha x)\sin(\beta x)\, dx \]
Answer: 0
Problem: The sequence $\{a_{n}\}_{n \geqslant 0}$ is defined by the initial conditions $a_{0}=1$ and $a_{1}=-4$, with the recurrence relation $a_{n+2}=-4a_{n+1}-7a_{n}$ for $n \geqslant 0$. Find the number of positive integer divisors of $a^2_{50} - a_{49}a_{51}$.
Answer: 51
Problem: Andrew writes down all of the prime numbers less than $50$. How many times does he write the digit $2$?
Answer: 3
Problem: A knight is placed at the origin of the Cartesian plane. Each turn, the knight moves in an L-shape (2 units parallel to one axis and 1 unit parallel to the other) to one of eight possible locations, chosen at random. After 2016 such turns, what is the expected value of the square of the distance of the knight from the origin?
Answer: 10080
Problem: Given a rectangle $ABCD$, where the segment $MA$ is perpendicular to the plane $ABC$. The lengths are as follows: $MB = 15$, $MC = 24$, and $MD = 20$. Determine the length of $MA$.
Answer: 7
Problem: For $a > 0$, find the minimum value of $\int_{0}^{\frac{1}{a}}(a^{3}+4x-a^{5}x^{2})e^{ax}\ dx.$
Answer: 4
Problem: Let $(x, y, z)$ be an ordered triplet of real numbers that satisfies the following system of equations:  
\[ \begin{align*} x + y^2 + z^4 &= 0, \\ y + z^2 + x^4 &= 0, \\ z + x^2 + y^4 &= 0. \end{align*} \]  
If $m$ is the minimum possible value of $\lfloor x^3 + y^3 + z^3 \rfloor$, find the modulo $2007$ residue of $m$.
Answer: 2006
Problem: Define a sequence $a_i$ as follows: \( a_1 = 181 \) and for \( i \ge 2 \), \( a_i = a_{i-1}^2 - 1 \) if \( a_{i-1} \) is odd and \( a_i = \frac{a_{i-1}}{2} \) if \( a_{i-1} \) is even. Find the least \( i \) such that \( a_i = 0 \).
Answer: 105
Problem: A $20 \times 20 \times 20$ block is cut up into 8000 non-overlapping unit cubes, and a number is assigned to each cube. It is known that in each column of 20 cubes parallel to any edge of the block, the sum of their numbers is equal to 1. The number assigned to one of the unit cubes is 10. Three $1 \times 20 \times 20$ slices parallel to the faces of the block contain this unit cube. Find the sum of all numbers of the cubes outside these slices.
Answer: 333
Problem: For what digit $d$ is the base $9$ numeral $7d35_9$ divisible by $8$?
Answer: 1
Problem: A real number $x$ satisfies $2 + \log_{25} x + \log_8 5 = 0$. Find $\log_2 x - (\log_8 5)^3 - (\log_{25} x)^3$.
Answer: 8
Problem: Let $\sigma_1 : \mathbb{N} \to \mathbb{N}$ be a function that takes a natural number $n$, and returns the sum of the positive integer divisors of $n$. For example, $\sigma_1(6) = 1 + 2 + 3 + 6 = 12$. What is the largest number $n$ such that $\sigma_1(n) = 1854$?
Answer: 1234
Problem: Suppose $a$, $b$, $c$, and $d$ are non-negative integers such that
\[(a+b+c+d)(a^2+b^2+c^2+d^2)^2=2023.\]
Find $a^3+b^3+c^3+d^3$.
Answer: 43
Problem: A cup with a volume of $8$ fluid ounces is filled at a rate of $0.5$ ounces per second. However, a hole at the bottom of the cup also drains it at a rate of $0.3$ ounces per second. Once the cup is full, how many ounces of water will have drained out of the cup?
Answer: 12
Problem: Evaluate the expression:
\[
\frac{(2 + 2)^2}{2^2} \cdot \frac{(3 + 3 + 3 + 3)^3}{(3 + 3 + 3)^3} \cdot \frac{(6 + 6 + 6 + 6 + 6 + 6)^6}{(6 + 6 + 6 + 6)^6}
\]
Answer: 108
Problem: Given a table with numbers $1, 2, \ldots, 18$, determine the minimal number of numbers that must be erased so that the sum of any two remaining numbers is not a perfect square of a positive integer.
Answer: 9
Problem: Let $\triangle ABC$ be an equilateral triangle with side length $s$ and $P$ a point in the interior of this triangle. Suppose that $PA$, $PB$, and $PC$ are the roots of the polynomial $t^3-18t^2+91t-89$. Then $s^2$ can be written in the form $m+\sqrt{n}$ where $m$ and $n$ are positive integers. Find $100m+n$.
Answer: 7208
Problem: Let $\{a_n\}_{n\geq 1}$ be an arithmetic sequence and $\{g_n\}_{n\geq 1}$ be a geometric sequence such that the first four terms of $\{a_n+g_n\}$ are $0$, $0$, $1$, and $0$, in that order. What is the $10$th term of $\{a_n+g_n\}$?
Answer: -54
Problem: Each of the squares in a $2 \times 2018$ grid is to be colored black or white such that in any $2 \times 2$ block, at least one of the four squares is white. Let $P$ be the number of ways of coloring the grid. Find the largest $k$ such that $3^k$ divides $P$.
Answer: 1009
Problem: A quadrilateral is given, where the lengths of two sides are $1$ and $4$. Additionally, a diagonal of length $2$ divides the quadrilateral into two isosceles triangles. Determine the perimeter of this quadrilateral.
Answer: 11
Problem: What is the maximum possible value for the sum of the squares of the roots of \(x^4 + ax^3 + bx^2 + cx + d\) where \(a\), \(b\), \(c\), and \(d\) are \(2\), \(0\), \(1\), and \(7\) in some order?
Answer: 49
Problem: Let $Q$ be a set of permutations of $1, 2, \ldots, 100$ such that for all $1 \leq a, b \leq 100$, $a$ can be found to the left of $b$ and adjacent to $b$ in at most one permutation in $Q$. Find the largest possible number of elements in $Q$.
Answer: 100
Problem: The non-zero natural number $n$ is a perfect square. By dividing $2023$ by $n$, we obtain the remainder $223 - \frac{3}{2} \cdot n$. Find the quotient of the division.
Answer: 14
Problem: Let $a = 3^{\frac{1}{223}} + 1$ and for all $n \geq 3$, let:
\[
f(n) = \binom{n}{0} a^{n-1} - \binom{n}{1} a^{n-2} + \binom{n}{2} a^{n-3} - \ldots + (-1)^{n-1} \binom{n}{n-1} a^0.
\]
Find $f(2007) + f(2008).$
Answer: 19683
Problem: Suppose that $f(x) = \frac{x}{x^2 - 2x + 2}$ and $g(x_1, x_2, \ldots , x_7) = f(x_1) + f(x_2) + \ldots + f(x_7)$. If $x_1, x_2, \ldots, x_7$ are non-negative real numbers with sum $5$, determine for how many tuples $(x_1, x_2, \ldots , x_7)$ does $g(x_1, x_2, \ldots , x_7)$ obtain its maximal value.
Answer: 1
Problem: How many positive integers $N$ less than $10^{1000}$ are such that $N$ has $x$ digits when written in base ten and $\frac{1}{N}$ has $x$ digits after the decimal point when written in base ten? For example, 20 has two digits and $\frac{1}{20} = 0.05$ has two digits after the decimal point, so $20$ is a valid $N$.
Answer: 3997
Problem: Let $a_1, a_2, \ldots$ and $b_1, b_2, \ldots$ be sequences such that $a_i b_i - a_i - b_i = 0$ and $a_{i+1} = \frac{2-a_i b_i}{1-b_i}$ for all $i \ge 1$. If $a_1 = 1 + \frac{1}{\sqrt[4]{2}}$, then what is $b_{6}$?
Answer: 257
Problem: Lynnelle took 10 tests in her math class at Stanford. Her score on each test was an integer from 0 through 100. She noticed that, for every four consecutive tests, her average score on those four tests was at most 47.5. What is the largest possible average score she could have on all 10 tests?
Answer: 57
Problem: We are given five watches which can be wound forward. What is the smallest sum of winding intervals that allows us to set them to the same time, regardless of their initial settings?
Answer: 24
Problem: The vertices of a regular nonagon are colored such that:
1. Adjacent vertices are different colors.
2. If 3 vertices form an equilateral triangle, they are all different colors.

Let $m$ be the minimum number of colors needed for a valid coloring, and $n$ be the total number of colorings using $m$ colors. Determine $mn$. (Assume each vertex is distinguishable.)
Answer: 54
Problem: A subset of $\{1, 2, 3, \ldots, 2017, 2018\}$ has the property that none of its members are 5 times another. What is the maximum number of elements that such a subset could have?
Answer: 1682
Problem: Let $x$, $y$, and $z$ be real numbers such that:
\[
12x - 9y^2 = 7 \\
6y - 9z^2 = -2 \\
12z - 9x^2 = 4
\]
Find the value of $6x^2 + 9y^2 + 12z^2$.
Answer: 9
Problem: The figure shows a convex polygon with nine vertices. The six diagonals drawn dissect the polygon into the following seven triangles: 
- $P_{0}P_{1}P_{3}$
- $P_{0}P_{3}P_{6}$
- $P_{0}P_{6}P_{7}$
- $P_{0}P_{7}P_{8}$
- $P_{1}P_{2}P_{3}$
- $P_{3}P_{4}P_{6}$
- $P_{4}P_{5}P_{6}$

In how many ways can these triangles be labeled with the names $\triangle_{1}$, $\triangle_{2}$, $\triangle_{3}$, $\triangle_{4}$, $\triangle_{5}$, $\triangle_{6}$, $\triangle_{7}$ such that $P_{i}$ is a vertex of triangle $\triangle_{i}$ for $i = 1, 2, 3, 4, 5, 6, 7$? Justify your answer.
Answer: 1
Problem: Two circles with radius $2$ and radius $4$ have a common center at $P$. Points $A$, $B$, and $C$ on the larger circle are the vertices of an equilateral triangle. Point $D$ is the intersection of the smaller circle and the line segment $PB$. Find the square of the area of triangle $ADC$.
Answer: 192
Problem: Mark writes the expression $\sqrt{d}$ for each positive divisor $d$ of $8!$ on the board. Rishabh simplifies each expression to the form $a \sqrt{b}$, where $a$ and $b$ are integers such that $b$ is not divisible by the square of a prime number. For example, $\sqrt{20}$, $\sqrt{16}$, and $\sqrt{6}$ simplify to $2\sqrt{5}$, $4\sqrt{1}$, and $1\sqrt{6}$, respectively. Compute the sum of $a+b$ across all expressions that Rishabh writes.
Answer: 3480
Problem: Karl likes the number $17$. His favorite polynomials are monic quadratics with integer coefficients such that $17$ is a root of the quadratic and the roots differ by no more than $17$. Compute the sum of the coefficients of all of Karl's favorite polynomials. (A monic quadratic is a quadratic polynomial whose $x^2$ term has a coefficient of $1$.)
Answer: 8960
Problem: Adi the Baller is shooting hoops, and makes a shot with probability $p$. He keeps shooting hoops until he misses. The value of $p$ that maximizes the chance that he makes between 35 and 69 (inclusive) buckets can be expressed as $\frac{1}{\sqrt[b]{a}}$ for a prime $a$ and positive integer $b$. Find $a+b$.
Answer: 37
Problem: How many pairs $(m, n)$ of positive integers with $m < n$ fulfill the equation $\frac{3}{2008} = \frac{1}{m} + \frac{1}{n}$?
Answer: 5
Problem: What is the number of ordered pairs $(A,B)$ where $A$ and $B$ are subsets of $\{1,2,..., 5\}$ such that neither $A \subseteq B$ nor $B \subseteq A$?
Answer: 570
Problem: Compute the base 10 value of $14641_{99}$.
Answer: 100000000
Problem: When two distinct digits are randomly chosen in $N=123456789$ and their places are swapped, one gets a new number $N'$ (for example, if 2 and 4 are swapped, then $N'=143256789$). The expected value of $N'$ is equal to $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Compute the remainder when $m+n$ is divided by $10^6$.
Answer: 962963
Problem: Find the largest value of $a^b$ such that the positive integers $a,b > 1$ satisfy:
\[ a^b b^a + a^b + b^a = 5329 \]
Answer: 81
Problem: Find the smallest positive integer $n$ for which the number \[ A_n = \prod_{k=1}^n \binom{k^2}{k} = \binom{1}{1} \binom{4}{2} \cdots \binom{n^2}{n} \] ends in the digit $0$ when written in base ten.
Answer: 4
Problem: There are real numbers $a, b, c, d$ such that for all $(x, y)$ satisfying $6y^2 = 2x^3 + 3x^2 + x$, if $x_1 = ax + b$ and $y_1 = cy + d$, then $y_1^2 = x_1^3 - 36x_1$. What is $a + b + c + d$?
Answer: 90
Problem: Find the greatest integer less than the number

$$1 + \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{3}} + \cdots + \frac{1}{\sqrt{1000000}}$$
Answer: 1998
Problem: Find the smallest positive $\alpha$ (in degrees) for which all the numbers $\cos{\alpha}, \cos{2\alpha}, \ldots, \cos{2^n\alpha}, \ldots$ are negative.
Answer: 120
Problem: **Problem Statement:**

One hundred friends, including Alice and Bob, live in several cities. Alice has determined the distance from her city to the city of each of the other 99 friends and totaled these 99 numbers. Alice’s total is \(1000\) km. Bob similarly totaled his distances to everyone else. What is the largest total that Bob could have obtained? 

*(Consider the cities as points on the plane; if two people live in the same city, the distance between their cities is considered zero).*
Answer: 99000
Problem: Determine the smallest non-negative integer $n$ such that \( \sqrt{(6n+11)(6n+14)(20n+19)} \in \mathbb{Q} \).
Answer: 219
Problem: Let $ABCD$ be a parallelogram. Let $G$ and $H$ be the feet of the altitudes from $A$ to $CD$ and $BC$, respectively. If $AD = 15$, $AG = 12$, and $AH = 16$, find the length of $AB$.
Answer: 20
Problem: Justin the robot is on a mission to rescue abandoned treasure from a minefield. To do this, he must travel from the point $(0, 0, 0)$ to $(4, 4, 4)$ in three-dimensional space, only taking one-unit steps in the positive $x$, $y$, or $z$-directions. However, the evil David anticipated Justin's arrival and has placed a mine at the point $(2, 2, 2)$. If at any point Justin is at most one unit away from this mine (in any direction), the mine detects his presence and explodes, thwarting Justin.

How many paths can Justin take to reach his destination safely?
Answer: 10350
Problem: What is the maximum number of subsets of size $5$, taken from the set $A = \{1, 2, 3, \ldots, 20\}$ such that any two of them share exactly one element?
Answer: 16
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: A sequence of distinct circles $\omega_1, \omega_2, \cdots$ is inscribed in the parabola $y=x^2$ such that $\omega_n$ and $\omega_{n+1}$ are tangent for all $n$. If $\omega_1$ has a diameter of 1 and touches the parabola at $(0,0)$, find the diameter of $\omega_{1998}$.
Answer: 3995
Problem: A snowman is built on a level plane by placing a ball of radius $6$ on top of a ball of radius $8$, which is on top of a ball of radius $10$, as shown. If the average height above the plane of a point in the snowman is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, find $m + n$.
Answer: 61
Problem: $DEB$ is a chord of a circle such that $DE=3$ and $EB=5$. Let $O$ be the center of the circle. Join $OE$ and extend $OE$ to cut the circle at $C$. Given $EC=1$, find the radius of the circle.
Answer: 8
Problem: In the middle of the school year, $40\%$ of Poolesville magnet students decided to transfer to the Blair magnet, and $5\%$ of the original Blair magnet students transferred to the Poolesville magnet. If the Blair magnet grew from $400$ students to $480$ students, how many students does the Poolesville magnet have after the transferring has occurred?
Answer: 170
Problem: For $n$ a positive integer, define $f_1(n) = n$. Then for $i$ a positive integer, define $f_{i+1}(n) = f_i(n)^{f_i(n)}$. Determine $f_{100}(75) \pmod{17}$. Justify your answer.
Answer: 1
Problem: There are $4$ pairs of men and women, and all $8$ people are arranged in a row such that in each pair, the woman is somewhere to the left of the man. How many such arrangements are there?
Answer: 2520
Problem: How many ways are there to color the $8$ regions of a three-set Venn Diagram with $3$ colors such that each color is used at least once? Two colorings are considered the same if one can be reached from the other by rotation and/or reflection.
Answer: 1248
Problem: The Dinky is a train connecting Princeton to the outside world. It runs on an odd schedule: the train arrives once every one-hour block at some uniformly random time (once at a random time between $9\text{am}$ and $10\text{am}$, once at a random time between $10\text{am}$ and $11\text{am}$, and so on). One day, Emilia arrives at the station, at some uniformly random time, and does not know the time. She expects to wait for $y$ minutes for the next train to arrive. After waiting for an hour, a train has still not come. She now expects to wait for $z$ minutes. Find $yz$.
Answer: 525
Problem: We delete the four corners of an $8 \times 8$ chessboard. How many ways are there to place eight non-attacking rooks on the remaining squares?
Answer: 21600
Problem: Let $A_1A_2A_3\ldots A_{12}$ be a regular dodecagon. Find the number of right triangles whose vertices are in the set $\{A_1, A_2, A_3, \ldots, A_{12}\}$.
Answer: 60
Problem: What is the sum of the prime factors of \(20!\)?
Answer: 77
Problem: Determine the sixth digit after the decimal point in the number $(\sqrt{1978} + \lfloor \sqrt{1978} \rfloor)^{20}$.
Answer: 9
Problem: Distinct integers $x, y, z$ satisfy the equation $(x-y)(y-z)(z-x) = x+y+z$. Determine the smallest possible value of $|x+y+z|$.
Answer: 54
Problem: Let the complex number $z = \cos\frac{1}{1000} + i \sin\frac{1}{1000}$. Find the smallest positive integer $n$ such that $z^n$ has an imaginary part that exceeds $\frac{1}{2}$.
Answer: 524
Problem: Find all prime numbers $p$ for which there exists an integer $n$ such that there are no integers $x, y$ satisfying $x^3 + y^3 \equiv n \pmod{p}$ (i.e., not all residues are the sum of two cubes).

For example, for $p = 7$, one could set $n = \pm 3$ since $x^3, y^3 \equiv 0, \pm 1 \pmod{7}$, thus $x^3 + y^3 \equiv 0, \pm 1, \pm 2 \pmod{7}$ only.
Answer: 7
Problem: Suppose that $P$ is a polynomial with integer coefficients such that $P(1) = 2$, $P(2) = 3$, and $P(3) = 2016$. If $N$ is the smallest possible positive value of $P(2016)$, find the remainder when $N$ is divided by $2016$.
Answer: 2013
Problem: Determine all real numbers $q$ for which the equation $x^4 - 40x^2 + q = 0$ has four real solutions that form an arithmetic progression.
Answer: 144
Problem: Camila creates a pattern to write the following numbers:
\[
2, 4 \\
5, 7, 9, 11 \\
12, 14, 16, 18, 20, 22 \\
23, 25, 27, 29, 31, 33, 35, 37 \\
\ldots
\]
Following the same pattern, what is the sum of the numbers in the tenth row?
Answer: 3840
Problem: How many numbers appear in both the arithmetic sequence $10, 16, 22, 28, \ldots, 1000$ and the arithmetic sequence $10, 21, 32, 43, \ldots, 1000$?
Answer: 16
Problem: Let $X$ be a point inside convex quadrilateral $ABCD$ such that $\angle AXB + \angle CXD = 180^{\circ}$. Given $AX = 14$, $BX = 11$, $CX = 5$, $DX = 10$, and $AB = CD$, find the sum of the areas of $\triangle AXB$ and $\triangle CXD$.
Answer: 90
Problem: In a circle, let $AB$ and $BC$ be chords, with $AB = \sqrt{3}$, $BC = 3\sqrt{3}$, and $\angle ABC = 60^\circ$. Find the length of the circle chord that bisects the angle $\angle ABC$.
Answer: 4
Problem: Evaluate $2000^3 - 1999 \cdot 2000^2 - 1999^2 \cdot 2000 + 1999^3$
Answer: 3999
Problem: Let $a_1$, $a_2$, \dots, $a_{2015}$ be a sequence of positive integers in $[1,100]$. Call a nonempty contiguous subsequence of this sequence "good" if the product of the integers in it leaves a remainder of $1$ when divided by $101$. In other words, it is a pair of integers $(x, y)$ such that $1 \le x \le y \le 2015$ and \[a_xa_{x+1}\dots a_{y-1}a_y \equiv 1 \pmod{101}. \] Find the minimum possible number of good subsequences across all possible $(a_i)$.
Answer: 19320