ChenDRAG/VeRL_math_validation huggingface dataset:
Problem: The diagram shows a regular pentagon $ABCDE$ and a square $ABFG$. Find the degree measure of $\angle FAD$.
Answer: 18
Problem: Let $ABC$ be a scalene triangle whose side lengths are positive integers. It is called stable if its three side lengths are multiples of 5, 80, and 112, respectively. What is the smallest possible side length that can appear in any stable triangle?
Answer: 20
Problem: Let $A$ and $B$ be two finite sets such that there are exactly $144$ sets which are subsets of $A$ or subsets of $B$. Find the number of elements in $A \cup B$.
Answer: 8
Problem: On equilateral $\triangle{ABC}$, point D lies on BC a distance 1 from B, point E lies on AC a distance 1 from C, and point F lies on AB a distance 1 from A. Segments AD, BE, and CF intersect in pairs at points G, H, and J, which are the vertices of another equilateral triangle. The area of $\triangle{ABC}$ is twice the area of $\triangle{GHJ}$. The side length of $\triangle{ABC}$ can be written as $\frac{r+\sqrt{s}}{t}$, where r, s, and t are relatively prime positive integers. Find $r + s + t$.
Answer: 30
Problem: Let $\sigma(n)$ represent the number of positive divisors of $n$. Define $s(n)$ as the number of positive divisors of $n+1$ such that for every divisor $a$, $a-1$ is also a divisor of $n$. Find the maximum value of $2s(n) - \sigma(n)$.
Answer: 2
Problem: The sequence of integers $a_1, a_2, \dots$ is defined as follows:
- $a_1 = 1$.
- For $n > 1$, $a_{n+1}$ is the smallest integer greater than $a_n$ such that $a_i + a_j \neq 3a_k$ for any $i, j, k$ from $\{1, 2, \dots, n+1\}$, where $i, j, k$ are not necessarily different.

Define $a_{2004}$.
Answer: 4507
Problem: For $n$ measured in degrees, let $T(n) = \cos^2(30^\circ -n) - \cos(30^\circ -n)\cos(30^\circ +n) + \cos^2(30^\circ +n)$. Evaluate $4\sum^{30}_{n=1} n \cdot T(n).$
Answer: 1395
Problem: Let $P(x)$ be a polynomial with positive integer coefficients and $P(0) > 1$. The product of all the coefficients is $47$ and the remainder when $P(x)$ is divided by $(x - 3)$ is $9887$. What is the degree of $P(x)$?
Answer: 8
Problem: Let $a_1$, $a_2$, $a_3$, $a_4$, and $a_5$ be real numbers satisfying
\begin{align*}
a_1a_2 + a_2a_3 + a_3a_4 + a_4a_5 + a_5a_1 & = 20,\\
a_1a_3 + a_2a_4 + a_3a_5 + a_4a_1 + a_5a_2 & = 22.
\end{align*}
Then the smallest possible value of $a_1^2 + a_2^2 + a_3^2 + a_4^2 + a_5^2$ can be expressed as $m + \sqrt{n}$, where $m$ and $n$ are positive integers. Compute $100m + n$.
Answer: 2105
Problem: Find the area of the $MNRK$ trapezoid with the lateral side $RK = 3$ if the distances from the vertices $M$ and $N$ to the line $RK$ are $5$ and $7$, respectively.
Answer: 18
Problem: Find the least positive integer $n$ such that the polynomial $P(X) = \sqrt{3} \cdot X^{n+1} - X^n - 1$ has at least one root of modulus $1$.
Answer: 10
Problem: Determine the minimal number of operations needed to calculate $x^{2006}$ using only multiplication and division of any two given or already obtained powers of the number $x$. For example, $x \cdot x = x^{2}$, $x^{2} \cdot x^{2} = x^{4}$, $x^{4} : x = x^{3}$, etc.
Answer: 14
Problem: Let set $S$ contain all positive integers that are one less than a perfect square. Find the sum of all powers of $2$ that can be expressed as the product of two (not necessarily distinct) members of $S$.
Answer: 64
Problem: Let $f(x,y)$ be a function defined for all pairs of nonnegative integers $(x, y)$, such that $f(0,k) = f(k,0) = 2^k$ and 
\[ f(a,b) + f(a+1,b+1) = f(a+1,b) + f(a,b+1) \]
for all nonnegative integers $a, b$. Determine the number of positive integers $n \leq 2016$ for which there exist two nonnegative integers $a, b$ such that $f(a,b) = n$.
Answer: 65
Problem: Suppose that the following equations hold for positive integers $x$, $y$, and $n$, where $n > 18$:
\[
\begin{align*}
x + 3y & \equiv 7 \pmod{n} \\
2x + 2y & \equiv 18 \pmod{n} \\
3x + y & \equiv 7 \pmod{n}
\end{align*}
\]
Compute the smallest nonnegative integer $a$ such that $2x \equiv a \pmod{n}$.
Answer: 20
Problem: Let $\triangle ABC$ 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 $\triangle ABC$ again at the point $M \neq A$. 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
Problem: Let $a$ and $b$ be positive integers such that $1 \leq a < b \leq 100$. If there exists a positive integer $k$ such that $ab \mid a^k + b^k$, we say that the pair $(a, b)$ is good. Determine the number of good pairs.
Answer: 96
Problem: Let $f$ be a function from the set $X = \{1,2, \dots, 10\}$ to itself. Call a partition $(S, T)$ of $X$ $f$-balanced if for all $s \in S$, we have $f(s) \in T$, and for all $t \in T$, we have $f(t) \in S$. 

A partition $(S, T)$ is a pair of subsets $S$ and $T$ of $X$ such that $S \cap T = \emptyset$ and $S \cup T = X$. Note that $(S, T)$ and $(T, S)$ are considered the same partition.

Let $g(f)$ be the number of $f$-balanced partitions, and let $m$ equal the maximum value of $g$ over all functions $f$ from $X$ to itself. If there are $k$ functions satisfying $g(f) = m$, determine $m+k$.
Answer: 961
Problem: Find the least natural number $n$ such that, if the set $\{1,2, \ldots, n\}$ is arbitrarily divided into two non-intersecting subsets, then one of the subsets contains 3 distinct numbers such that the product of two of them equals the third.
Answer: 96
Problem: Let $m$ and $n$ be positive integers such that $m^4 - n^4 = 3439$. What is the value of $mn$?
Answer: 90
Problem: In a triangle $ABC$, $\angle BAC = 90^{\circ}$. Let $D$ be the point on $BC$ such that $AB + BD = AC + CD$. Suppose $BD : DC = 2:1$. If $\frac{AC}{AB} = \frac{m + \sqrt{p}}{n}$, where $m, n$ are relatively prime positive integers and $p$ is a prime number, determine the value of $m+n+p$.
Answer: 34
Problem: How many sequences of real numbers $a_1, a_2, \ldots, a_9$ satisfy 
\[ |a_1 - 1| = |a_2 - a_1| = \cdots = |a_9 - a_8| = |1 - a_9| = 1? \]
Answer: 252
Problem: Find the number of 5-digit numbers that each contains the block '15' and is divisible by 15.
Answer: 479
Problem: Let $p=2017$ be a prime. Find the remainder when \[ \left\lfloor \frac{1^p}{p} \right\rfloor + \left\lfloor \frac{2^p}{p} \right\rfloor + \left\lfloor \frac{3^p}{p} \right\rfloor + \cdots + \left\lfloor \frac{2015^p}{p} \right\rfloor \] is divided by $p$. Here $\lfloor \cdot \rfloor$ denotes the greatest integer function.
Answer: 1010
Problem: Start with a three-digit positive integer $A$. Obtain $B$ by interchanging the two leftmost digits of $A$. Obtain $C$ by doubling $B$. Obtain $D$ by subtracting $500$ from $C$. Given that $A + B + C + D = 2014$, find $A$.
Answer: 344
Problem: Five students take a test where any integer score from $0$ to $100$ is possible. What is the largest possible difference between the median and the mean of the scores?
Answer: 40
Problem: Let $C$ denote the curve $y^2 = \frac{x(x+1)(2x+1)}{6}$. The points $(\frac{1}{2}, a)$, $(b, c)$, and $(24, d)$ lie on $C$ and are collinear, and $ad < 0$. Given that $b$ and $c$ are rational numbers, find $100b^2 + c^2$.
Answer: 101
Problem: Leticia has a $9 \times 9$ board. She says that two squares are "friends" if they share a side, if they are at opposite ends of the same row, or if they are at opposite ends of the same column. Every square has $4$ friends on the board. Leticia will paint every square one of three colors: green, blue, or red. In each square, a number will be written based on the following rules:

- If the square is green, write the number of red friends plus twice the number of blue friends.
- If the square is red, write the number of blue friends plus twice the number of green friends.
- If the square is blue, write the number of green friends plus twice the number of red friends.

Considering that Leticia can choose the coloring of the squares on the board, find the maximum possible value she can obtain when she sums the numbers in all the squares.
Answer: 486
Problem: Let $S$ be the set of all positive integers from 1 through 1000 that are not perfect squares. What is the length of the longest, non-constant, arithmetic sequence that consists of elements of $S$?
Answer: 333
Problem: Find the area of the figure on the coordinate plane bounded by the straight lines $x = 0$, $x = 2$, and the graphs of the functions $y = \sqrt{x^3 + 1}$ and $y = -\sqrt[3]{x^2 + 2x}$.
Answer: 6
Problem: How many ordered pairs of integers $(x, y)$ satisfy the equation: 

$$2x^2 + y^2 + xy = 2(x + y)$$
Answer: 4
Problem: Compute the smallest positive integer $n$ such that $2016^n$ does not divide $2016!$.
Answer: 335
Problem: Find the sum of all three-digit numbers (written in base $10$) such that the leading digit is the sum of the other two digits. Express your answer in base $10$.
Answer: 34815
Problem: Find the number of positive integers $n$ such that the highest power of $7$ dividing $n!$ is $8$.
Answer: 7
Problem: Let $a$ and $b$ be nonzero real numbers such that $\frac{1}{3a} + \frac{1}{b} = 2011$ and $\frac{1}{a} + \frac{1}{3b} = 1$. What is the quotient when $a + b$ is divided by $ab$?
Answer: 1509
Problem: Consider the function $f(x) = 5x^4 - 12x^3 + 30x^2 - 12x + 5$. Let $f(x_1) = p$, where $x_1$ and $p$ are non-negative integers, and $p$ is prime. Find with proof the largest possible value of $p$.
Answer: 5
Problem: Find the largest positive integer $n$ not divisible by $10$ which is a multiple of each of the numbers obtained by deleting two consecutive digits (neither of them in the first or last position) of $n$.
Answer: 9999
Problem: Ryan chooses five subsets $S_1, S_2, S_3, S_4, S_5$ of the set $\{1, 2, 3, 4, 5, 6, 7\}$ such that $|S_1| = 1$, $|S_2| = 2$, $|S_3| = 3$, $|S_4| = 4$, and $|S_5| = 5$. Moreover, for all $1 \le i < j \le 5$, either $S_i \cap S_j = S_i$ or $S_i \cap S_j = \emptyset$ (in other words, the intersection of $S_i$ and $S_j$ is either $S_i$ or the empty set). In how many ways can Ryan select the sets?
Answer: 11760
Problem: There is a positive integer $s$ such that there are $s$ solutions to the equation
\[ 64\sin^2(2x) + \tan^2(x) + \cot^2(x) = 46 \]
in the interval \( (0, \frac{\pi}{2}) \), all of the form \( \frac{m_k}{n_k}\pi \), where $m_k$ and $n_k$ are relatively prime positive integers, for $k = 1, 2, 3, \ldots, s$. Find \((m_1 + n_1) + (m_2 + n_2) + (m_3 + n_3) + \cdots + (m_s + n_s)\).
Answer: 100
Problem: Let $f(x)$ be a continuous function over the real numbers such that for every integer $n$, $f(n) = n^2$ and $f(x)$ is linear over the interval $[n, n + 1]$. There exists a unique two-variable polynomial $g$ such that $g(x, \lfloor x \rfloor) = f(x)$ for all $x$. Compute $g(20, 23)$. Here, $\lfloor x \rfloor$ is defined as the greatest integer less than or equal to $x$. For example, $\lfloor 2 \rfloor = 2$ and $\lfloor -3.5 \rfloor = -4$.
Answer: 388
Problem: Find the greatest positive integer $n$ such that $n$ is divisible by all positive integers whose cube is not greater than $n$.
Answer: 420
Problem: Consider all pairs of points $(a,b,c)$ and $(d,e,f)$ in the 3-D coordinate system with the condition $ad + be + cf = -2023$. What is the least positive integer that can be the distance between such a pair of points?
Answer: 90
Problem: Find the smallest integer $n \geq 1$ such that the equation:
$$a^2 + b^2 + c^2 - nd^2 = 0$$
has $(0, 0, 0, 0)$ as the unique solution.
Answer: 7
Problem: Three people, John, Macky, and Rik, play a game of passing a basketball from one to another. Find the number of ways of passing the ball starting with Macky and reaching Macky again at the end of the seventh pass.
Answer: 42
Problem: Let $P(x)$ be a real quadratic trinomial such that for all $x \in \mathbb{R}$, the inequality $P(x^3 + x) \geq P(x^2 + 1)$ holds. Find the sum of the roots of $P(x)$.
Answer: 4
Problem: As in the following diagram, square $ABCD$ and square $CEFG$ are placed side by side (i.e. $C$ is between $B$ and $E$ and $G$ is between $C$ and $D$). If $CE = 14$, $AB > 14$, compute the minimal area of $\triangle AEG$. 

```asy
size(120); defaultpen(linewidth(0.7)+fontsize(10)); 
pair D2(real x, real y) {
 pair P = (x,y);
 dot(P,linewidth(3)); return P;
}
int big = 30, small = 14;
filldraw((0,big)--(big+small,0)--(big,small)--cycle, rgb(0.9,0.5,0.5));
draw(scale(big)*unitsquare); draw(shift(big,0)*scale(small)*unitsquare);
label("$A$",D2(0,big),NW);
label("$B$",D2(0,0),SW);
label("$C$",D2(big,0),SW);
label("$D$",D2(big,big),N);
label("$E$",D2(big+small,0),SE);
label("$F$",D2(big+small,small),NE);
label("$G$",D2(big,small),NE);
```

Note: Unnecessary information like author or edit history has been removed.
Answer: 98
Problem: Find the smallest possible value of $n$ such that $n+2$ people can stand inside or on the border of a regular $n$-gon with side length $6$ feet where each pair of people are at least $6$ feet apart.
Answer: 10
Problem: The sequence of nonnegative integers $F_0, F_1, F_2, \dots$ is defined recursively as $F_0 = 0$, $F_1 = 1$, and $F_{n+2} = F_{n+1} + F_n$ for all integers $n \geq 0$. Let $d$ be the largest positive integer such that, for all integers $n \geq 0$, $d$ divides $F_{n+2020} - F_n$. Compute the remainder when $d$ is divided by $1001$.
Answer: 638
Problem: How many ordered pairs of integers \((a,b)\) satisfy all of the following inequalities? 

\[ \begin{align*} 
& a^2 + b^2 < 16 \\
& a^2 + b^2 < 8a \\
& a^2 + b^2 < 8b 
\end{align*} \]
Answer: 6
Problem: In triangle $ABC$, $AB = 100$, $BC = 120$, and $CA = 140$. Points $D$ and $F$ lie on $\overline{BC}$ and $\overline{AB}$, respectively, such that $BD = 90$ and $AF = 60$. Point $E$ is an arbitrary point on $\overline{AC}$. Denote the intersection of $\overline{BE}$ and $\overline{CF}$ as $K$, the intersection of $\overline{AD}$ and $\overline{CF}$ as $L$, and the intersection of $\overline{AD}$ and $\overline{BE}$ as $M$. If $[KLM] = [AME] + [BKF] + [CLD]$, where $[X]$ denotes the area of region $X$, compute $CE$.
Answer: 91
Problem: Several machines were working in the workshop. After reconstruction, the number of machines decreased, and the percentage by which the number of machines decreased turned out to be equal to the number of remaining machines. What was the smallest number of machines that could have been in the workshop before the reconstruction?
Answer: 25
Problem: The function $f(n)$ is defined on the positive integers and takes non-negative integer values. It is given that $f(2) = 0$, $f(3) > 0$, and $f(9999) = 3333$. For all positive integers $m$ and $n$, the following condition holds:
\[ f(m+n) - f(m) - f(n) = 0 \text{ or } 1. \]
Determine the value of $f(1982)$.
Answer: 660
Problem: The \textit{arithmetic derivative} $D(n)$ of a positive integer $n$ is defined via the following rules:
\begin{itemize}
  \item $D(1) = 0$;
  \item $D(p) = 1$ for all primes $p$;
  \item $D(ab) = D(a)b + aD(b)$ for all positive integers $a$ and $b$.
\end{itemize}
Find the sum of all positive integers $n$ below $1000$ satisfying $D(n) = n$.
Answer: 31
Problem: Let $f(n)$ denote the sum of the distinct positive integer divisors of $n$. Evaluate:
\[
f(1) + f(2) + f(3) + f(4) + f(5) + f(6) + f(7) + f(8) + f(9).
\]
Answer: 69
Problem: Let $ABC$ be a triangle with $AB = 20$ and $AC = 18$. $E$ is on segment $AC$ and $F$ is on segment $AB$ such that $AE = AF = 8$. Let $BE$ and $CF$ intersect at $G$. Given that $AEGF$ is cyclic, then $BC = m\sqrt{n}$ for positive integers $m$ and $n$ such that $n$ is not divisible by the square of any prime. Compute $100m + n$.
Answer: 305
Problem: After Jennifer walked $r$ percent of the way from her home to the store, she turned around and walked home, got on her bicycle, and bicycled to the store and back home. Jennifer bicycles two and a half times faster than she walks. Find the largest value of $r$ so that returning home for her bicycle was not slower than her walking all the way to and from the store without her bicycle.
Answer: 60
Problem: There are $n$ sticks which have distinct integer lengths. Suppose that it's possible to form a non-degenerate triangle from any $3$ distinct sticks among them. It's also known that there are sticks of lengths $5$ and $12$ among them. What is the largest possible value of $n$ under such conditions?
Answer: 6
Problem: If Percy rolls a fair six-sided die until he rolls a $5$, what is his expected number of rolls, given that all of his rolls are prime?
Answer: 3
Problem: Consider a triangle $ABC$ and let $M$ be the midpoint of the side $BC$. Suppose $\angle MAC = \angle ABC$ and $\angle BAM = 105^{\circ}$. Find the measure of $\angle ABC$.
Answer: 30
Problem: What is the maximum number of distinct integers in a row such that the sum of any 11 consecutive integers is either 100 or 101?
Answer: 22
Problem: Let $\triangle ABC$ have $\angle ABC=67^{\circ}$. Point $X$ is chosen such that $AB = XC$, $\angle XAC=32^\circ$, and $\angle XCA=35^\circ$. Compute $\angle BAC$ in degrees.
Answer: 81
Problem: Start by writing the integers $1, 2, 4, 6$ on the blackboard. At each step, write the smallest positive integer $n$ that satisfies both of the following properties on the board:
1. $n$ is larger than any integer currently on the board.
2. $n$ cannot be written as the sum of two distinct integers already on the board.

Find the $100$-th integer that you write on the board. Recall that at the beginning, there are already $4$ integers on the board.
Answer: 388
Problem: Compute the number of nonnegative integer triples \((x, y, z)\) which satisfy the inequality:
\[ 4x + 2y + z \leq 36 \]
Answer: 1330
Problem: How many pairs of positive integers $(a, b)$ satisfy $\frac{1}{a} + \frac{1}{b} = \frac{1}{2004}$?
Answer: 45
Problem: In parallelogram $ABCD$, the longer side is twice the shorter side. Let $XYZW$ be the quadrilateral formed by the internal bisectors of the angles of $ABCD$. If the area of $XYZW$ is $10$, find the area of $ABCD$.
Answer: 40
Problem: Concentric circles $\Omega_1$ and $\Omega_2$ with radii $1$ and $100$, respectively, are drawn with center $O$. Points $A$ and $B$ are chosen independently at random on the circumferences of $\Omega_1$ and $\Omega_2$, respectively. Denote by $\ell$ the tangent line to $\Omega_1$ passing through $A$, and denote by $P$ the reflection of $B$ across $\ell$. Compute the expected value of $OP^2$.
Answer: 10004
Problem: Find the three-digit number that has the greatest number of different divisors.
Answer: 840
Problem: The corners of a $2$-dimensional room in the shape of an isosceles right triangle are labeled $A$, $B$, $C$ where $AB = BC$. Walls $BC$ and $CA$ are mirrors. A laser is shot from $A$, hits off each of the mirrors once, and lands at a point $X$ on $AB$. Let $Y$ be the point where the laser hits off $AC$. If $\frac{AB}{AX} = 64$, and $\frac{CA}{AY} = \frac{p}{q}$ for coprime positive integers $p$ and $q$, find $p + q$.
Answer: 131
Problem: Let $S_0 = \varnothing$ denote the empty set, and define $S_n = \{ S_0, S_1, \dots, S_{n-1} \}$ for every positive integer $n$. Find the number of elements in the set
\[ (S_{10} \cap S_{20}) \cup (S_{30} \cap S_{40}). \]
Answer: 30
Problem: What is the largest positive integer $n < 1000$ for which there is a positive integer $m$ satisfying \[ \text{lcm}(m,n) = 3m \times \gcd(m,n)? \]
Answer: 972
Problem: Three rods of lengths $1396$, $1439$, and $2018$ millimeters have been hinged from one tip on the ground. What is the smallest value for the radius of the circle passing through the other three tips of the rods in millimeters?
Answer: 311
Problem: In Prime Land, there are seven major cities, labeled $C_0$, $C_1$, \dots, $C_6$. For convenience, we let $C_{n+7} = C_n$ for each $n=0,1,\dots,6$; i.e., we take the indices modulo $7$. Al initially starts at city $C_0$.

Each minute for ten minutes, Al flips a fair coin. If the coin lands heads, and he is at city $C_k$, he moves to city $C_{2k}$; otherwise, he moves to city $C_{2k+1}$. If the probability that Al is back at city $C_0$ after $10$ moves is $\frac{m}{1024}$, find $m$.
Answer: 147
Problem: The degree measures of the six interior angles of a convex hexagon form an arithmetic sequence (not necessarily in cyclic order). The common difference of this arithmetic sequence can be any real number in the open interval $(-D, D)$. Compute the greatest possible value of $D$.
Answer: 24
Problem: The function $f : \mathbb{R} \to \mathbb{R}$ satisfies the equation \( f(x^2) f''(x) = f'(x) f'(x^2) \) for all real numbers \( x \). Given that \( f(1) = 1 \) and \( f'''(1) = 8 \), determine the value of \( f'(1) + f''(1) \).
Answer: 6
Problem: The value of $21!$ is $51,090,942,171,abc,440,000$, where $a$, $b$, and $c$ are digits. What is the value of $100a + 10b + c$?
Answer: 709
Problem: A cube $ABCDA'B'C'D'$ is given with an edge of length $2$, with vertices marked as in the figure. The point $K$ is the center of the edge $AB$. The plane containing the points $B', D', K$ intersects the edge $AD$ at point $L$. Calculate the volume of the pyramid with apex $A$ and base the quadrilateral $D'B'KL$.
Answer: 1
Problem: Let $r$, $s$, and $t$ be the roots of the polynomial $x^3 + 6x^2 + 7x + 8$. Calculate the expression:
\[
(r^2 + s + t)(s^2 + t + r)(t^2 + r + s)
\]
Answer: 1100
Problem: How many zeros does $101^{100} - 1$ end with?
Answer: 4
Problem: Find the number of integer solutions to $||x| - 2020| < 5$.
Answer: 18
Problem: Find the last eight digits of the binary representation of $27^{1986}$.
Answer: 11011001
Problem: The roots of the equation $x^5 - 180x^4 + Ax^3 + Bx^2 + Cx + D = 0$ are in geometric progression. The sum of their reciprocals is $20$. Compute $|D|$.
Answer: 243
Problem: The figure on the right shows the ellipse \( \frac{(x-19)^2}{19} + \frac{(y-98)^2}{98} = 1998 \).

Let \( R_1, R_2, R_3, \) and \( R_4 \) denote those areas within the ellipse that are in the first, second, third, and fourth quadrants, respectively. Determine the value of \( R_1 - R_2 + R_3 - R_4 \).
Answer: 7448
Problem: Let $x \neq y$ be positive reals satisfying $x^3 + 2013y = y^3 + 2013x$, and let $M = (\sqrt{3} + 1)x + 2y$. Determine the maximum possible value of $M^2$.
Answer: 16104
Problem: Let $\triangle ABC$ be a triangle with $AB = 5$, $AC = 4$, and $BC = 6$. The internal angle bisector of $\angle C$ intersects the side $AB$ at point $D$. Points $M$ and $N$ are taken on sides $BC$ and $AC$, respectively, such that $DM \parallel AC$ and $DN \parallel BC$. If $(MN)^2 = \frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers, what is the sum of the digits of $|p - q|$?
Answer: 2
Problem: In how many different orders can the characters $P \ U \ M \ \alpha \ C$ be arranged such that the $M$ is to the left of the $\alpha$ and the $\alpha$ is to the left of the $C$?
Answer: 20
Problem: Kevin is trying to solve an economics question which has six steps. At each step, he has a probability $p$ of making a sign error. Let $q$ be the probability that Kevin makes an even number of sign errors (thus answering the question correctly!). For how many values of $0 \le p \le 1$ is it true that $p+q=1$?
Answer: 2
Problem: Find $ \lim_{x\rightarrow 0} \frac{1}{x} \int_{-x}^x (t\sin 2006 t + 2007t + 1004)\ dt$.
Answer: 2008
Problem: Let $AB$ and $CD$ be two parallel chords in a circle with radius $5$ such that the center $O$ lies between these chords. Suppose $AB = 6$ and $CD = 8$. Further, suppose that the area of the part of the circle lying between the chords $AB$ and $CD$ is given by \( \frac{m\pi + n}{k} \), where $m$, $n$, and $k$ are positive integers with \( \gcd(m, n, k) = 1 \). What is the value of $m + n + k$?
Answer: 75
Problem: Find the largest prime that divides $1 \cdot 2 \cdot 3 + 2 \cdot 3 \cdot 4 + \cdots + 44 \cdot 45 \cdot 46$.
Answer: 47
Problem: For lunch, Lamy, Botan, Nene, and Polka each choose one of three options: a hot dog, a slice of pizza, or a hamburger. Lamy and Botan choose different items, and Nene and Polka choose the same item. In how many ways could they choose their items?
Answer: 18
Problem: Find the number of nonnegative integers $n$ for which $(n^2 - 3n + 1)^2 + 1$ is a prime number.
Answer: 4
Problem: Positive integers $x, y, z$ satisfy $xy + z = 160$. Compute the smallest possible value of $x + yz$.
Answer: 50