ChenDRAG/VeRL_math_validation huggingface dataset:
Problem: Compute the sum of all positive integers $n$ such that $n^n$ has 325 positive integer divisors. For example, $4^4=256$ has 9 positive integer divisors: 1, 2, 4, 8, 16, 32, 64, 128, 256.
Answer: 93
Problem: Points $E$ and $F$ lie on diagonal $\overline{AC}$ of square $ABCD$ with side length $24$, such that $AE = CF = 3\sqrt{2}$. An ellipse with foci at $E$ and $F$ is tangent to the sides of the square. Find the sum of the distances from any point on the ellipse to the two foci.
Answer: 30
Problem: Let $A = \frac{1}{6}((\log_2(3))^3-(\log_2(6))^3-(\log_2(12))^3+(\log_2(24))^3)$. Compute $2^A$.
Answer: 72
Problem: \( \triangle ABC \) has side lengths \( AB=20 \), \( BC=15 \), and \( CA=7 \). Let the altitudes of \( \triangle ABC \) be \( AD \), \( BE \), and \( CF \). What is the distance between the orthocenter (intersection of the altitudes) of \( \triangle ABC \) and the incenter of \( \triangle DEF \)?
Answer: 15
Problem: A box of strawberries, containing $12$ strawberries total, costs $2$. A box of blueberries, containing $48$ blueberries total, costs $3$. Suppose that for $12$, Sareen can either buy $m$ strawberries total or $n$ blueberries total. Find $n - m$.
Answer: 120
Problem: Determine the number of integer pairs \((x, y)\) such that \((|x| - 2)^2 + (|y| - 2)^2 < 5\).
Answer: 48
Problem: For any set $S$, let $P(S)$ be its power set, the set of all its subsets. Consider all sets $A$ of 2015 arbitrary finite sets. Let $N$ be the maximum possible number of ordered pairs $(S,T)$ such that $S \in P(A)$, $T \in P(P(A))$, $S \in T$, and $S \subseteq T$. Note that by convention, a set may never contain itself. Find the remainder when $N$ is divided by 1000.
Answer: 872
Problem: If $p$, $q$, and $r$ are nonzero integers satisfying $p^2+q^2 = r^2$, compute the smallest possible value of $(p+q+r)^2$.
Answer: 4
Problem: We are given a $5 \times 5$ square grid, divided into $1 \times 1$ tiles. Two tiles are called **linked** if they lie in the same row or column, and the distance between their centers is 2 or 3. For example, in the picture, the gray tiles are the ones linked to the red tile.

![Grid Image](https://i.imgur.com/JVTQ9wB.png)

Sammy wants to mark as many tiles in the grid as possible, such that no two of them are linked. What is the maximal number of tiles he can mark?
Answer: 10
Problem: In the land of Nyemo, the unit of currency is called a quack. The citizens use coins that are worth $1$, $5$, $25$, and $125$ quacks. How many ways can someone pay off $125$ quacks using these coins?
Answer: 82
Problem: Three circles $C_i$ are given in the plane: $C_1$ has diameter $AB$ of length $1$; $C_2$ is concentric and has diameter $k$ ($1 < k < 3$); $C_3$ has center $A$ and diameter $2k$. We regard $k$ as fixed. Consider all straight line segments $XY$ which have one endpoint $X$ on $C_2$, one endpoint $Y$ on $C_3$, and contain the point $B$. For what ratio $\frac{XB}{BY}$ will the segment $XY$ have minimal length?
Answer: 1
Problem: In the diagram, the circle has a radius of $\sqrt{7}$ and center $O$. Points $A$, $B$, and $C$ are on the circle. If $\angle BOC = 120^\circ$ and $AC = AB + 1$, determine the length of $AB$.
Answer: 4
Problem: Find the total number of integer solutions of the equation $x^5 - y^2 = 4$.
Answer: 0
Problem: Consider the polynomials:
\[ P(x) = (x + \sqrt{2})(x^2 - 2x + 2) \]
\[ Q(x) = (x - \sqrt{2})(x^2 + 2x + 2) \]
\[ R(x) = (x^2 + 2)(x^8 + 16) \]
Find the coefficient of \( x^4 \) in \( P(x) \cdot Q(x) \cdot R(x) \).
Answer: 0
Problem: While taking the SAT, you become distracted by your own answer sheet. Because you are not bound to the College Board's limiting rules, you realize that there are actually $32$ ways to mark your answer for each question, because you could fight the system and bubble in multiple letters at once: for example, you could mark $AB$, or $AC$, or $ABD$, or even $ABCDE$, or nothing at all!

You begin to wonder how many ways you could mark off the 10 questions you haven't yet answered. To increase the challenge, you wonder how many ways you could mark off the rest of your answer sheet without ever marking the same letter twice in a row. (For example, if $ABD$ is marked for one question, $AC$ cannot be marked for the next one because $A$ would be marked twice in a row.) If the number of ways to do this can be expressed in the form $2^m p^n$, where $m,n > 1$ are integers and $p$ is a prime, compute $100m+n+p$.
Answer: 2013
Problem: Alice the ant starts at vertex $A$ of regular hexagon $ABCDEF$ and moves either right or left each move with equal probability. After $35$ moves, what is the probability that she is on either vertex $A$ or $C$?
Answer: 0
Problem: There are $128$ numbered seats arranged around a circle in a palaestra. The first person to enter the place would sit on seat number $1$. Since a contagious disease is infecting the people of the city, each person who enters the palaestra would sit on a seat whose distance is the longest to the nearest occupied seat. If there are several such seats, the newly entered person would sit on the seat with the smallest number. What is the number of the seat on which the $39$th person sits?
Answer: 27
Problem: There are $5$ vertices labeled $1, 2, 3, 4, 5$. For any two pairs of vertices $u, v$, the edge $uv$ is drawn with probability $\frac{1}{2}$. If the probability that the resulting graph is a tree is given by $\frac{p}{q}$ where $p$ and $q$ are coprime, then find the value of $q^{1/10} + p$.
Answer: 127
Problem: Let $r$ be a nonzero real number. The values of $z$ which satisfy the equation
\[ r^4z^4 + (10r^6-2r^2)z^2-16r^5z+(9r^8+10r^4+1) = 0 \]
are plotted on the complex plane (i.e., using the real part of each root as the x-coordinate and the imaginary part as the y-coordinate). Show that the area of the convex quadrilateral with these points as vertices is independent of $r$, and find this area.
Answer: 8
Problem: Let the rational number $\frac{p}{q}$ be closest to but not equal to $\frac{22}{7}$ among all rational numbers with denominator $< 100$. What is the value of $p - 3q$?
Answer: 14
Problem: Let $x, y, z \in \mathbb{R}$ and the equation $7x^2 + 7y^2 + 7z^2 + 9xyz = 12$ holds. Find the minimum value of $x^2 + y^2 + z^2$, which can be expressed as $\frac{a}{b}$, where $a, b \in \mathbb{Z}$ and $\gcd(a, b) = 1$. Determine the value of $a + b$.
Answer: 7
Problem: In a triangle, the ratio of the interior angles is $1 : 5 : 6$, and the longest side has length $12$. What is the length of the altitude (height) of the triangle that is perpendicular to the longest side?
Answer: 3
Problem: For real numbers $x$, let 
$$f(x) = \frac{4^x}{25^{x+1}} + \frac{5^x}{2^{x+1}}.$$ 
Then, 
$$f\left(\frac{1}{1 - \log_{10} 4}\right) = \frac{m}{n},$$ 
where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
Answer: 15001
Problem: Find the remainder when $712!$ is divided by $719$.
Answer: 718
Problem: Suppose the side lengths of triangle $ABC$ are the roots of the polynomial $x^3 - 27x^2 + 222x - 540$. What is the product of its inradius and circumradius?
Answer: 10
Problem: For each positive integer $n$, let $S_n$ denote the set of positive integers $k$ such that $n^k - 1$ is divisible by $2006$. Define the function $P(n)$ by the rule:

$$P(n) = \begin{cases} \min(s)_{s \in S_n} & \text{if } S_n \neq \emptyset, \\ 0 & \text{otherwise}. \end{cases}$$

Let $d$ be the least upper bound of $\{P(1), P(2), P(3), \ldots\}$, and let $m$ be the number of integers $i$ such that $1 \leq i \leq 2006$ and $P(i) = d$. Compute the value of $d + m$.
Answer: 912
Problem: How many reorderings of \(2, 3, 4, 5, 6\) have the property that every pair of adjacent numbers are relatively prime?
Answer: 4
Problem: For some positive integers $m > n$, the quantities $a = \text{lcm}(m, n)$ and $b = \gcd(m, n)$ satisfy $a = 30b$. If $m - n$ divides $a$, then what is the value of $\frac{m + n}{b}$?
Answer: 11
Problem: If $x^3 - 3\sqrt{3} x^2 + 9x - 3\sqrt{3} - 64 = 0$, find the value of $x^6 - 8x^5 + 13x^4 - 5x^3 + 49x^2 - 137x + 2015$.
Answer: 1898
Problem: Suppose $ABCD$ is a rectangle whose diagonals meet at $E$. The perimeter of triangle $ABE$ is $10\pi$ and the perimeter of triangle $ADE$ is $n$. Compute the number of possible integer values of $n$.
Answer: 47
Problem: The numbers in the sequence 101, 104, 109, 116, $\dots$ are of the form $a_n = 100 + n^2$, where $n = 1, 2, 3, \dots$. For each $n$, let $d_n$ be the greatest common divisor of $a_n$ and $a_{n + 1}$. Find the maximum value of $d_n$ as $n$ ranges through the positive integers.
Answer: 401
Problem: When $4 \cos \theta - 3 \sin \theta = \frac{13}{3},$ it follows that $7 \cos 2\theta - 24 \sin 2\theta = \frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$
Answer: 122
Problem: A sequence $\{a_n\}$ satisfies the following conditions:  
- $a_0 = 0$  
- $a_1 = a_2 = 1$  
- $a_{3n} = a_n$  
- $a_{3n+1} = a_{3n+2} = a_n + 1$  
for all $n \geq 1$. Compute $a_{2021}$.
Answer: 6
Problem: Euler's formula states that for a convex polyhedron with $V$ vertices, $E$ edges, and $F$ faces, $V - E + F = 2$. A particular convex polyhedron has 32 faces, each of which is either a triangle or a pentagon. At each of its $V$ vertices, $T$ triangular faces and $P$ pentagonal faces meet. What is the value of $100P + 10T + V$?
Answer: 250
Problem: Let $f(x) = x^3 + 3x - 1$ have roots $a, b, c$. Given that \(\frac{1}{a^3 + b^3} + \frac{1}{b^3 + c^3} + \frac{1}{c^3 + a^3}\) can be written as \(\frac{m}{n}\), where \(m, n\) are positive integers and \(\gcd(m, n) = 1\), find \(100m + n\).
Answer: 3989
Problem: The George Washington Bridge is $2016$ meters long. Sally is standing on the bridge, $1010$ meters from its left end. Each step, she either moves $1$ meter to the left or $1$ meter to the right, each with probability $\frac{1}{2}$. What is the expected number of steps she will take to reach an end of the bridge?
Answer: 1016060
Problem: Compute the number of positive integers less than or equal to $10000$ which are relatively prime to $2014$.
Answer: 4648
Problem: There are $2022$ stones on a table. At the start of the game, Teacher Tseng will choose a positive integer $m$ and let Ming and LTF play a game. LTF is the first to move, and he can remove at most $m$ stones on his round. Then the two players take turns removing stones. Each round, they must remove at least one stone, and they cannot remove more than twice the amount of stones the last person removed. The player unable to move loses. Find the smallest positive integer $m$ such that LTF has a winning strategy.
Answer: 1
Problem: Let $a$, $b$, $c$ be the distinct roots of the polynomial $P(x) = x^3 - 10x^2 + x - 2015$. The cubic polynomial $Q(x)$ is monic and has distinct roots $bc-a^2$, $ca-b^2$, $ab-c^2$. What is the sum of the coefficients of $Q$?
Answer: 2015000
Problem: Find the number of solutions to the equation: 

$$6\{x\}^3 + \{x\}^2 + \{x\} + 2x = 2018.$$ 

With $\{x\}$, we denote the fractional part of the number $x$.
Answer: 5
Problem: Alfred and Bonnie play a game where they take turns tossing a fair coin. The winner of the game is the first person to obtain a head. Alfred and Bonnie play this game multiple times with the rule that the loser of a game goes first in the next game. Suppose Alfred goes first in the first game, and the probability that he wins the sixth game is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What are the last three digits of $m + n$?
Answer: 93
Problem: One morning, each member of Manjul’s family drank an $8$-ounce mixture of coffee and milk. The amounts of coffee and milk varied from cup to cup, but were never zero. Manjul drank $\frac{1}{7}$ of the total amount of milk and $\frac{2}{17}$ of the total amount of coffee. How many people are there in Manjul’s family?
Answer: 8
Problem: Bob, a spherical person, is floating around peacefully when Dave the giant orange fish launches him straight up at 23 m/s with his tail. If Bob has a density of 100 $\text{kg/m}^3$, let $f(r)$ denote how far underwater his center of mass plunges once he lands, assuming his center of mass was at water level when he's launched up. Find $\lim_{r \to 0} \left(f(r)\right)$. Express your answer in meters and round to the nearest integer. Assume the density of water is 1000 $\text{kg/m}^3$.
Answer: 3
Problem: Find the smallest positive integer $a$ such that $x^4 + a^2$ is not prime for any integer $x$.
Answer: 8
Problem: Broady The Boar is playing a board game consisting of a circle with $2021$ points on it, labeled $0$, $1$, $2$, \ldots, $2020$ in that order clockwise. Broady is rolling a $2020$-sided die which randomly produces a whole number between $1$ and $2020$, inclusive.

Broady starts at the point labeled $0$. After each dice roll, Broady moves up the same number of points as the number rolled (point $2020$ is followed by point $0$). For example, if they are at $0$ and roll a $5$, they end up at $5$. If they are at $2019$ and roll a $3$, they end up at $1$.

Broady continues rolling until they return to the point labeled $0$. What is the expected number of times they roll the dice?
Answer: 2021
Problem: Find the greatest positive integer $n$ such that $3^n$ divides $70! + 71! + 72!$.
Answer: 36
Problem: Determine the largest natural number $m$ such that for each non-negative real numbers $a_1 \ge a_2 \ge \ldots \ge a_{2014} \ge 0$, the following inequality holds: $$\frac{a_1 + a_2 + \ldots + a_m}{m} \ge \sqrt{\frac{a_1^2 + a_2^2 + \ldots + a_{2014}^2}{2014}}.$$
Answer: 44
Problem: Counting from the right end, what is the $2500$th digit of $10000!$?
Answer: 2
Problem: Determine the last digit of $5^5 + 6^6 + 7^7 + 8^8 + 9^9$.
Answer: 9
Problem: While there do not exist pairwise distinct real numbers $a,b,c$ satisfying $a^2+b^2+c^2 = ab+bc+ca$, there do exist complex numbers with that property. Let $a,b,c$ be complex numbers such that $a^2+b^2+c^2 = ab+bc+ca$ and $|a+b+c| = 21$. Given that $|a-b| = 2\sqrt{3}$, $|a| = 3\sqrt{3}$, compute $|b|^2+|c|^2$.
Answer: 132
Problem: Find the last three digits of 
\[2008^{2007^{\cdot^{\cdot^{\cdot ^{2^1}}}}}.\]
Answer: 8
Problem: Wu starts with exactly one coin. Each year, Wu flips every coin he has at once. For each heads he flips, Wu receives a coin, and for each tails he flips, Wu loses a coin. He continues this process each year until he has $0$ coins, at which point he stops. The probability that Wu will stop after exactly five years can be expressed as $\frac{a}{2^b}$, where $a$ and $b$ are positive integers with $a$ being odd. Find $a+b$.
Answer: 71622400
Problem: Three non-zero real numbers $a$, $b$, and $c$ satisfy the conditions $a + b + c = 0$ and $a^4 + b^4 + c^4 = 128$. Determine all possible values of $ab + bc + ca$.
Answer: -8
Problem: Let $a$, $b$, $c$ be positive real numbers for which:
\[
  \frac{5}{a} = b+c, \quad
  \frac{10}{b} = c+a, \quad \text{and} \quad
  \frac{13}{c} = a+b.
\]
If $a+b+c = \frac{m}{n}$ for relatively prime positive integers $m$ and $n$, compute $m+n$.
Answer: 55
Problem: There are five dots arranged in a line from left to right. Each of the dots is colored from one of five colors such that no three consecutive dots are all the same color. How many ways are there to color the dots?
Answer: 2800
Problem: An acute triangle $ABC$ has side lengths $a$, $b$, $c$ such that $a$, $b$, $c$ form an arithmetic sequence. Given that the area of triangle $ABC$ is an integer, what is the smallest value of its perimeter?
Answer: 42
Problem: The three-digit number $999$ has a special property: It is divisible by $27$, and its digit sum is also divisible by $27$. The four-digit number $5778$ also has this property, as it is divisible by $27$ and its digit sum is also divisible by $27$. How many four-digit numbers have this property?
Answer: 75
Problem: Let $ABC$ be a triangle with side lengths $AB=6$, $AC=7$, and $BC=8$. Let $H$ be the orthocenter of $\triangle ABC$ and $H'$ be the reflection of $H$ across the midpoint $M$ of $BC$. The expression $\frac{[ABH']}{[ACH']}$ can be written as $\frac{p}{q}$. Find $p+q$.
Answer: 251
Problem: How many ordered triples of integers $(x, y, z)$ are there such that $x^2 + y^2 + z^2 = 34$?
Answer: 48
Problem: For a bijective function $g : \mathbb{R} \to \mathbb{R}$, we say that a function $f : \mathbb{R} \to \mathbb{R}$ is its superinverse if it satisfies the following identity: 
\[(f \circ g)(x) = g^{-1}(x)\]
where $g^{-1}$ is the inverse of $g$. Given:
\[g(x) = x^3 + 9x^2 + 27x + 81\]
and $f$ is its superinverse, find:
\[|f(-289)|\]
Answer: 7
Problem: Triangle $\triangle ABC$ has $\angle{A}=90^\circ$ with $BC=12$. Square $BCDE$ is drawn such that $A$ is in its interior. The line through $A$ tangent to the circumcircle of $\triangle ABC$ intersects $CD$ and $BE$ at $P$ and $Q$, respectively. If $PA=4\cdot QA$, and the area of $\triangle ABC$ can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$, then compute $m+n$.
Answer: 149
Problem: At a recent math contest, Evan was asked to find $2^{2016} \pmod{p}$ for a given prime number $p$ with $100 < p < 500$. Evan has forgotten what the prime $p$ was, but still remembers how he solved it:

- Evan first tried taking $2016$ modulo $p - 1$, but got a value $e$ larger than $100$.
- However, Evan noted that $e - \frac{1}{2}(p - 1) = 21$, and then realized the answer was $-2^{21} \pmod{p}$.

What was the prime $p$?
Answer: 211
Problem: Farmer James invents a new currency such that for every positive integer $n \leq 6$, there exists an $n$-coin worth $n!$ cents. Furthermore, he has exactly $n$ copies of each $n$-coin. An integer $k$ is said to be *nice* if Farmer James can make $k$ cents using at least one copy of each type of coin. How many positive integers less than 2018 are nice?
Answer: 210
Problem: A board with dimensions $2018 \times 2018$ is divided into unit cells of size $1 \times 1$. In some cells of the board, black chips are placed, and in others, white chips are placed (each cell contains at most one chip). 

First, we remove all black chips from columns that contain white chips. Then, we remove all white chips from rows that contain black chips. 

Let $W$ be the number of remaining white chips, and $B$ be the number of remaining black chips on the board. Define $A = \min\{W, B\}$. Determine the maximum value of $A$.
Answer: 1018081
Problem: Find how many integer values $3 \le n \le 99$ satisfy that the polynomial $x^2 + x + 1$ divides $x^{2^n} + x + 1$.
Answer: 49
Problem: A chicken breeder went to check the price per chick he charged the previous year. He found an invoice, half erased, which read: \(72\) chickens sold for \(*679*\) escudos" (the digits of the units and tens of thousands were illegible). What price did each chick sell for last year?
Answer: 511
Problem: Find all positive integers $n$ such that $n$ is a perfect number and $\varphi(n)$ is a power of $2$.
Answer: 6
Problem: Let $ABCD$ be a square with side length $2$. Let $M$ and $N$ be the midpoints of $\overline{BC}$ and $\overline{CD}$ respectively, and let $X$ and $Y$ be the feet of the perpendiculars from $A$ to $\overline{MD}$ and $\overline{NB}$, also respectively. The square of the length of segment $\overline{XY}$ can be written in the form $\frac{p}{q}$ where $p$ and $q$ are positive relatively prime integers. What is $100p+q$?
Answer: 3225
Problem: Let $S$ be the set of natural numbers that cannot be written as the sum of three squares. Legendre's three-square theorem states that $S$ consists of precisely the integers of the form $4^a(8b+7)$ where $a$ and $b$ are nonnegative integers. Find the smallest $n \in \mathbb{N}$ such that $n$ and $n+1$ are both in $S$.
Answer: 111
Problem: A trapezoid has side lengths as indicated in the figure, with the sides of lengths $11$ and $36$ being parallel. Calculate the area of the trapezoid. 

![Trapezoid Image](https://1.bp.blogspot.com/-5PKrqDG37X4/XzcJtCyUv8I/AAAAAAAAMY0/tB0FObJUJdcTlAJc4n6YNEaVIDfQ91-eQCLcBGAsYHQ/s0/1995%2BMohr%2Bp1.png)
Answer: 564
Problem: Four friends purchased a ball. The first friend paid half of the ball's total price. The second friend contributed one-third of the total amount paid by the other three friends. The third friend paid a quarter of the sum paid by the other three friends. The fourth friend paid $5. How much did the ball cost?
Answer: 100
Problem: Find the last two digits of $\binom{200}{100}$. Express the answer as an integer between $0$ and $99$. (e.g., if the last two digits are $05$, just write $5$).
Answer: 20
Problem: Find the minimum value of \( S = |x + 1| + |x + 5| + |x + 14| + |x + 97| + |x + 1920| \).
Answer: 2011
Problem: Find \( \sum_{i=1}^{2016} i(i+1)(i+2) \pmod{2018} \).
Answer: 0
Problem: Determine the largest positive integer that is a divisor of all three of \( A=2^{2010}\times3^{2010} \), \( B=3^{2010}\times5^{2010} \), and \( C=5^{2010}\times2^{2010} \).
Answer: 1
Problem: $AB$ is the diameter of circle $O$. A random point $P$ is selected on $O$ such that $AP = 4$ and $BP = 3$. Points $C$ and $D$ are drawn on circle $O$ such that $OC$ bisects $AP$ and $OD$ bisects $BP$. What is the degree measure of $\angle COD$?
Answer: 90
Problem: If \((2^x - 4^x) + (2^{-x} - 4^{-x}) = 3\), find the numerical value of the expression 
\[
(8^x + 3\cdot 2^x) + (8^{-x} + 3\cdot 2^{-x}).
\]
Answer: -1
Problem: Beto plays the following game with his computer: initially, the computer randomly picks $30$ integers from $1$ to $2015$, and Beto writes them on a chalkboard (there may be repeated numbers). On each turn, Beto chooses a positive integer $k$ and some of the numbers written on the chalkboard, and subtracts $k$ from each of the chosen numbers, with the condition that the resulting numbers remain non-negative. The objective of the game is to reduce all $30$ numbers to $0$, in which case the game ends. Find the minimal number $n$ such that, regardless of which numbers the computer chooses, Beto can end the game in at most $n$ turns.
Answer: 11
Problem: Suppose that $a$, $b$, $c$, and $d$ are real numbers such that $a+b+c+d=8$. Compute the minimum possible value of 
\[20(a^2+b^2+c^2+d^2)-\sum_{\text{sym}}a^3b,\]
where the sum is over all $12$ symmetric terms.
Answer: 112
Problem: Xavier takes a permutation of the numbers $1$ through $2011$ at random, where each permutation has an equal probability of being selected. He then cuts the permutation into increasing contiguous subsequences, such that each subsequence is as long as possible. Compute the expected number of such subsequences.

**Clarification**: An increasing contiguous subsequence is an increasing subsequence all of whose terms are adjacent in the original sequence. For example, $1,3,4,5,2$ has two maximal increasing contiguous subsequences: $(1,3,4,5)$ and $(2)$.
Answer: 1006
Problem: Three lines $m$, $n$, and $\ell$ lie in a plane such that no two are parallel. Lines $m$ and $n$ meet at an acute angle of $14^{\circ}$, and lines $m$ and $\ell$ meet at an acute angle of $20^{\circ}$. Find, in degrees, the sum of all possible acute angles formed by lines $n$ and $\ell$.
Answer: 40
Problem: How many sequences of words (not necessarily grammatically correct) have the property that the first word has one letter, each word can be obtained by inserting a letter somewhere in the previous word, and the final word is $\text{CCAMT}$? Here are examples of possible sequences:

- $C, CA, CAM, CCAM, CCAMT$.
- $A, AT, CAT, CAMT, CCAMT$.
Answer: 60
Problem: How many of the first $1000$ positive integers can be written as the sum of finitely many distinct numbers from the sequence $3^0$, $3^1$, $3^2$, $\ldots$?
Answer: 105
Problem: The quadrilateral $ABCD$ is an isosceles trapezoid with $AB = CD = 1$, $BC = 2$, and $DA = 1 + \sqrt{3}$. What is the measure of $\angle ACD$ in degrees?
Answer: 90
Problem: Find the sum of $pq$ for all pairs of prime numbers $p \leq q$ such that $pq \mid 15(p-1)(q-1)$.
Answer: 31
Problem: Find the least positive integer $n$ such that there exists a polynomial \[ P(x) = a_{2n}x^{2n} + a_{2n-1}x^{2n-1} + \dots + a_1x + a_0 \] with real coefficients satisfying the following properties:
- For $i = 0, 1, \dots, 2n$, it holds that $2014 \leq a_i \leq 2015$.
- There exists a real number $\xi$ such that $P(\xi) = 0$.
Answer: 2014