ChenDRAG/VeRL_math_validation huggingface dataset:
Problem: Ali defines a pronunciation of any sequence of English letters to be a partition of those letters into substrings such that each substring contains at least one vowel. For example, $\text{A } | \text{ THEN } | \text{ A}$, $\text{ATH } | \text{ E } | \text{ NA}$, $\text{ATHENA}$, and $\text{AT } | \text{ HEN } | \text{ A}$ are all pronunciations of the sequence $\text{ATHENA}$. How many distinct pronunciations does $\text{YALEMATHCOMP}$ have? (Y is not a vowel.)
Answer: 45
Problem: Suppose you have $27$ identical unit cubes colored such that $3$ faces adjacent to a vertex are red and the other $3$ are colored blue. Suppose further that you assemble these $27$ cubes randomly into a larger cube with $3$ cubes to an edge (in particular, the orientation of each cube is random). The probability that the entire cube is one solid color can be written as $\frac{1}{2^n}$ for some positive integer $n$. Find $n$.
Answer: 53
Problem: Suppose $x$, $y$, and $z$ are nonzero complex numbers such that $(x+y+z)(x^2+y^2+z^2)=x^3+y^3+z^3$. Compute $$(x+y+z)\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\right).$$
Answer: 3
Problem: Given the sequence $1, 2, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 1, \ldots$, find $n$ such that the sum of the first $n$ terms is $2008$ or $2009$.
Answer: 1026
Problem: Let $x=\sqrt{37-20\sqrt{3}}$. Find the value of \(\frac{x^4-9x^3+5x^2-7x+68}{x^2-10x+19}\).
Answer: 7
Problem: Compute the only element of the set \[ \{1, 2, 3, 4, \ldots\} \cap \left\{\frac{404}{r^2-4} \;\bigg| \; r \in \mathbb{Q} \setminus \{-2, 2\}\right\}. \]
Answer: 2500
Problem: In $\triangle ABC$, three lines are drawn parallel to side $BC$, dividing the altitude of the triangle into four equal parts. If the area of the second largest part is $35$, what is the area of the whole $\triangle ABC$?
Answer: 112
Problem: One test is a multiple choice test with $5$ questions, each having $4$ options. There are $2000$ candidates, each choosing only one answer for each question. Find the smallest possible integer $n$ such that a student's answer sheet, labeled $n$, has the following property: The answer sheet contains four sheets, and any two of these four sheets have exactly the same three answers.
Answer: 5
Problem: How many possible values are there for the sum $a + b + c + d$ if $a$, $b$, $c$, and $d$ are positive integers and $abcd = 2007$?
Answer: 4
Problem: Twelve friends participate in a tennis tournament, where each friend plays exactly one game against each of the other eleven friends. The winner of each game receives one point, while the loser receives zero points. There are no draws. The final points for the participants are denoted as $B_1, B_2, \ldots, B_{12}$. Determine the largest possible value of the sum $\Sigma_3 = B_1^3 + B_2^3 + \ldots + B_{12}^3$.
Answer: 4356
Problem: Find the greatest value of the positive integer $x$, such that the number \( A = 2^{182} + 4^x + 8^{700} \) is a perfect square.
Answer: 2008
Problem: Anthony denotes in sequence all positive integers divisible by $2$. Bertha denotes in sequence all positive integers divisible by $3$. Claire denotes in sequence all positive integers divisible by $4$. Orderly Dora denotes all numbers written by the other three. She puts them in order by size and does not repeat a number. What is the $2017^{th}$ number in her list?
Answer: 3026
Problem: Let $S_n$ be the sum of reciprocal values of non-zero digits of all positive integers up to (and including) $n$. For instance, $S_{13} = \frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \frac{1}{6} + \frac{1}{7} + \frac{1}{8} + \frac{1}{9} + \frac{1}{1} + \frac{1}{1} + \frac{1}{1} + \frac{1}{1} + \frac{1}{2} + \frac{1}{1} + \frac{1}{3}$.

Find the least positive integer $k$ making the number $k! \cdot S_{2016}$ an integer.
Answer: 7
Problem: Circles $P$, $Q$, and $R$ are externally tangent to one another. The external tangent of circles $P$ and $Q$ that does not intersect circle $R$ intersects circle $P$ at point $P_Q$ and circle $Q$ at point $Q_P$. Points $Q_R$, $R_Q$, $R_P$, and $P_R$ are defined similarly.

If the radius of circle $Q$ is $4$ and the line segment $\overline{Q_PP_Q}$ is parallel to $\overline{R_QQ_R}$, compute the length of $R_PP_R$.
Answer: 8
Problem: Joel is rolling a 6-sided die. After his first roll, he can choose to re-roll the die up to 2 more times. If he rerolls strategically to maximize the expected value of the final value the die lands on, the expected value of the final value the die lands on can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$?
Answer: 17
Problem: On a $5 \times 5$ grid, we randomly place two \emph{cars}, each occupying a single cell and randomly facing one of the four cardinal directions. It is given that the two cars do not start in the same cell. In a \emph{move}, one chooses a car and shifts it one cell forward. The probability that there exists a sequence of moves such that, afterward, both cars occupy the same cell is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Compute $100m + n$.
Answer: 1148
Problem: David draws a $2 \times 2$ grid of squares in chalk on the sidewalk outside NIMO HQ. He then draws one arrow in each square, each pointing in one of the four cardinal directions (north, south, east, west) parallel to the sides of the grid. In how many ways can David draw his arrows such that no two of the arrows are pointing at each other?
Answer: 194
Problem: Call a set of positive integers "conspiratorial" if no three of them are pairwise relatively prime. What is the largest number of elements in any "conspiratorial" subset of the integers $1$ to $16$?
Answer: 11
Problem: Nine distinct light bulbs are arranged in a circle, each initially turned off. Determine the number of ways to turn on some of these bulbs such that no four consecutive bulbs remain off at the same time.
Answer: 367
Problem: The positive integer $n$ has the following property: if the last three digits of $n$ are removed, the number becomes $\sqrt[3]{n}$. Find $n$.
Answer: 32768
Problem: In the diagram below, square $ABCD$ with side length 23 is cut into nine rectangles by two lines parallel to $\overline{AB}$ and two lines parallel to $\overline{BC}$. The areas of four of these rectangles are indicated in the diagram. Compute the largest possible value for the area of the central rectangle.

[asy]
size(250);
 defaultpen (linewidth (0.7) + fontsize (10));
 draw ((0,0)--(23,0)--(23,23)--(0,23)--cycle);
 label("$A$", (0,23), NW);
 label("$B$", (23, 23), NE);
 label("$C$", (23,0), SE);
 label("$D$", (0,0), SW);
 draw((0,6)--(23,6));
 draw((0,19)--(23,19));
 draw((5,0)--(5,23));
 draw((12,0)--(12,23));
 label("13", (17/2, 21));
 label("111",(35/2,25/2));
 label("37",(17/2,3));
 label("123",(2.5,12.5));
[/asy]
Answer: 180
Problem: Let $n = 2^{19} \times 3^{12}$. Let $M$ denote the number of positive divisors of $n^2$ which are less than $n$ but would not divide $n$. What is the number formed by taking the last two digits of $M$ (in the same order)?
Answer: 28
Problem: Consider non-negative real numbers $a, b, c$ satisfying the condition $a^2 + b^2 + c^2 = 2$. Find the maximum value of the following expression:
\[
P = \frac{\sqrt{b^2 + c^2}}{3-a} + \frac{\sqrt{c^2 + a^2}}{3-b} + a + b - 2022c
\]
Answer: 3
Problem: How many integer pairs $(a, b)$ with $1 < a, b \le 2015$ are there such that $\log_a b$ is an integer?
Answer: 2082
Problem: A T-tetromino is formed by adjoining three unit squares to form a $1 \times 3$ rectangle, and adjoining on top of the middle square a fourth unit square. Determine the least number of unit squares that must be removed from a $202 \times 202$ grid so that it can be tiled using T-tetrominoes.
Answer: 4
Problem: Let $ABC$ be a triangle with $AB = 13$, $BC = 14$, and $CA = 15$. Points $P$, $Q$, and $R$ are chosen on segments $BC$, $CA$, and $AB$, respectively, such that triangles $AQR$, $BPR$, and $CPQ$ have the same perimeter, which is $\frac{4}{5}$ of the perimeter of $\triangle PQR$. What is the perimeter of $\triangle PQR$?
Answer: 30
Problem: The parabolas $y = x^2 + 15x + 32$ and $x = y^2 + 49y + 593$ meet at one point $(x_0, y_0)$. Find $x_0 + y_0$.
Answer: -31
Problem: Let $A = (0,0)$, $B=(-1,-1)$, $C=(x,y)$, and $D=(x+1,y)$, where $x > y$ are positive integers. Suppose points $A$, $B$, $C$, and $D$ lie on a circle with radius $r$. Denote by $r_1$ and $r_2$ the smallest and second smallest possible values of $r$. Compute $r_1^2 + r_2^2$.
Answer: 2523
Problem: Let $A_1A_2A_3 \ldots A_{21}$ be a 21-sided regular polygon inscribed in a circle with center $O$. How many triangles $A_iA_jA_k$, $1 \leq i < j < k \leq 21$, contain the center point $O$ in their interior?
Answer: 385
Problem: For what largest $n$ are there $n$ seven-digit numbers that are successive members of one geometric progression?
Answer: 11
Problem: The sum of the following seven numbers is exactly 19:  
\[a_1=2.56,\ a_2=2.61,\ a_3=2.65,\ a_4=2.71,\ a_5=2.79,\ a_6=2.82,\ a_7=2.86.\]  
It is desired to replace each $a_i$ by an integer approximation $A_i$, where $1 \le i \le 7$, so that the sum of the $A_i$'s is also 19 and the maximum of the "errors" $|A_i - a_i|$, denoted as $M$, is minimized. For this minimum $M$, what is $100M$?
Answer: 61
Problem: We say a positive integer $n$ is \emph{metallic} if there is no prime of the form $m^2-n$. What is the sum of the three smallest metallic integers?
Answer: 165
Problem: Let $ABC$ be a triangle. Let the $A$-altitude meet $BC$ at $D$, the $B$-altitude meet $AC$ at $E$, and let $T \neq A$ be the point on the circumcircle of $\triangle ABC$ such that $AT \parallel BC$. Given that $D, E, T$ are collinear, if $BD = 3$ and $AD = 4$, then the area of $\triangle ABC$ can be written as $a + \sqrt{b}$, where $a$ and $b$ are positive integers. What is $a + b$?
Answer: 112
Problem: Triangle $ABC$ is inscribed in a unit circle $\omega$. Let $H$ be its orthocenter and $D$ be the foot of the perpendicular from $A$ to $BC$. Let $\triangle XYZ$ be the triangle formed by drawing the tangents to $\omega$ at $A$, $B$, $C$. If $\overline{AH} = \overline{HD}$ and the side lengths of $\triangle XYZ$ form an arithmetic sequence, the area of $\triangle ABC$ can be expressed in the form $\frac{p}{q}$ for relatively prime positive integers $p$, $q$. What is $p + q$?
Answer: 11
Problem: A trapezoid has side lengths $10, 10, 10$, and $22$. Each side of the trapezoid is the diameter of a semicircle. The two semicircles on the parallel sides of the trapezoid face outside the trapezoid, while the other two semicircles face inside, as shown in the figure.

The region bounded by these four semicircles has an area expressed as $m + n\pi$, where $m$ and $n$ are positive integers. Find $m + n$.
Answer: 176
Problem: Find the number of positive integers $n$ such that $\sqrt{n} + \sqrt{n + 1} < 11$.
Answer: 29
Problem: The coefficients of the polynomial $P(x)$ are nonnegative integers, each less than 100. Given that $P(10) = 331633$ and $P(-10) = 273373$, compute $P(1)$.
Answer: 100
Problem: A child lines up $2020^2$ pieces of bricks in a row and removes bricks whose positions are square numbers (i.e., the 1st, 4th, 9th, 16th, ... bricks). Then, he lines up the remaining bricks again and removes those that are in a 'square position'. This process is repeated until the number of bricks remaining drops below $250$. How many bricks remain in the end?
Answer: 240
Problem: Let $a_1, a_2, a_3, \ldots$ be an infinite sequence where for all positive integers $i$, $a_i$ is chosen to be a random positive integer between $1$ and $2016$, inclusive. Let $S$ be the set of all positive integers $k$ such that for all positive integers $j < k$, $a_j \neq a_k$. (So $1 \in S$; $2 \in S$ if and only if $a_1 \neq a_2$; $3 \in S$ if and only if $a_1 \neq a_3$ and $a_2 \neq a_3$; and so on.) In simplest form, let $\frac{p}{q}$ be the expected number of positive integers $m$ such that $m$ and $m+1$ are in $S$. Compute $pq$.
Answer: 4030
Problem: The number of proper subsets of the set $\left\{ x \mid -1 \leq \log_{\frac{1}{x}} 10 < -\frac{1}{2}, x \in \mathbb{Z}_{\geq 0} \right\}$ is ________.
Answer: 127
Problem: Aileen plays badminton where she and her opponent stand on opposite sides of a net and attempt to bat a birdie back and forth over the net. A player wins a point if their opponent fails to bat the birdie over the net. When Aileen is the server (the first player to try to hit the birdie over the net), she wins a point with probability $\frac{9}{10}$. Each time Aileen successfully bats the birdie over the net, her opponent, independent of all previous hits, returns the birdie with probability $\frac{3}{4}$. Each time Aileen bats the birdie, independent of all previous hits, she returns the birdie with probability $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
Answer: 73
Problem: Count the sum of all four-digit positive integers that contain only odd digits in their decimal representation. Note that the digits can only be 1, 3, 5, 7, and 9.
Answer: 3471875
Problem: Triangle $ABC$ is isosceles, and $\angle ABC = x^{\circ}$. If the sum of the possible measures of $\angle BAC$ is $240^{\circ}$, find $x$.
Answer: 20
Problem: 13 LHS Students attend the LHS Math Team tryouts. The students are numbered $1, 2, \ldots, 13$. Their scores are $s_1, s_2, \ldots, s_{13}$, respectively. There are 5 problems on the tryout, each of which is given a weight, labeled $w_1, w_2, \ldots, w_5$. Each score $s_i$ is equal to the sum of the weights of all problems solved by student $i$. On the other hand, each weight $w_j$ is assigned to be $\frac{1}{\sum_{s_i}}$, where the sum is over all the scores of students who solved problem $j$. (If nobody solved a problem, the score doesn't matter). If the largest possible average score of the students can be expressed in the form $\frac{\sqrt{a}}{b}$, where $a$ is square-free, find $a+b$.
Answer: 78
Problem: Let $ABC$ be an acute triangle with $\angle ABC = 60^{\circ}$. Suppose points $D$ and $E$ are on lines $AB$ and $CB$, respectively, such that $\triangle CDB$ and $\triangle AEB$ are equilateral triangles. Given that the positive difference between the perimeters of $\triangle CDB$ and $\triangle AEB$ is $60$ and $DE = 45$, what is the value of $AB \cdot BC?$
Answer: 1625
Problem: The edges and diagonals of convex pentagon $ABCDE$ are all colored either red or blue. How many ways are there to color the segments such that there is exactly one monochromatic triangle with vertices among $A$, $B$, $C$, $D$, $E$; that is, triangles whose edges are all the same color?
Answer: 260
Problem: Let $f : \mathbb{N} \to \mathbb{R}$ be a function, satisfying the following condition: for every integer $n > 1$, there exists a prime divisor $p$ of $n$ such that $f(n) = f \left(\frac{n}{p}\right) - f(p)$. If $f(2^{2007}) + f(3^{2008}) + f(5^{2009}) = 2006$, determine the value of $f(2007^2) + f(2008^3) + f(2009^5)$.
Answer: 9
Problem: Let $ABCD$ be a rectangle. Circles with diameters $AB$ and $CD$ meet at points $P$ and $Q$ inside the rectangle such that $P$ is closer to segment $BC$ than $Q$. Let $M$ and $N$ be the midpoints of segments $AB$ and $CD$. If $\angle MPN = 40^\circ$, find the degree measure of $\angle BPC$.
Answer: 160
Problem: The integer $n$ has exactly six positive divisors, which are $1 < a < b < c < d < n$. Let $k = a - 1$. If the $k$-th divisor (according to the above ordering) of $n$ is equal to $(1 + a + b)b$, find the highest possible value of $n$.
Answer: 2009
Problem: Let $\triangle ABC$ have median $CM$ ($M \in AB$) and circumcenter $O$. The circumcircle of $\triangle AMO$ bisects $CM$. Determine the least possible perimeter of $\triangle ABC$ if it has integer side lengths.
Answer: 24
Problem: For every four-digit number, calculate the product of its four digits. Then, sum all of these products together. What is the result?
Answer: 4100625
Problem: A cubic block with dimensions $n \times n \times n$ is composed of $1 \times 1 \times 1$ unit cubes. Determine the smallest value of $n$ such that, after removing the outer two layers of unit cubes, more than half of the original unit cubes remain.
Answer: 20
Problem: Six distinct positive integers are randomly chosen between $1$ and $2011$, inclusive. The probability that some pair of the six chosen integers has a difference that is a multiple of $5$ is $n$ percent. Find $n$.
Answer: 100
Problem: We define a positive integer as **Noble** if it meets the following criteria:  
1. It is a composite number.  
2. It is not divisible by any prime number greater than $20$.  
3. It is not divisible by any perfect cube greater than $1$.  

Determine the total number of distinct Noble numbers.
Answer: 6552
Problem: Let $a_1$, $a_2$, \ldots$ be an infinite sequence of integers such that $0 \le a_k \le k$ for every positive integer $k$ and such that \[ 2017 = \sum_{k = 1}^\infty a_k \cdot k! \, . \] What is the value of the infinite series $\sum_{k = 1}^\infty a_k$?
Answer: 11
Problem: In triangle $ABC$, let $M$ be the midpoint of $BC$, $H$ be the orthocenter, and $O$ be the circumcenter. Let $N$ be the reflection of $M$ over $H$. Suppose that $OA = ON = 11$ and $OH = 7$. Compute $BC^2$.
Answer: 288
Problem: In an isosceles triangle $ABC$ with base $BC$, let $M$ be the midpoint of $BC$. Let $P$ be the intersection of the circumcircle of $\triangle ACM$ with the circle centered at $B$ passing through $M$, such that $P \neq M$. If $\angle BPC = 135^\circ$, then $\frac{CP}{AP}$ can be expressed as $a + \sqrt{b}$ for positive integers $a$ and $b$, where $b$ is not divisible by the square of any prime. Find $a + b$.
Answer: 3
Problem: The three-digit number $n = CCA$ (in base $10$), where $C \neq A$, is divisible by $14$. How many possible values for $n$ are there?
Answer: 7
Problem: Consider the Harmonic Table:
\[
\begin{array}{c@{\hspace{15pt}}c@{\hspace{15pt}}c@{\hspace{15pt}}c@{\hspace{15pt}}c@{\hspace{15pt}}c@{\hspace{15pt}}c}
&&&1&&&\\
&&\tfrac{1}{2}&&\tfrac{1}{2}&&\\
&\tfrac{1}{3}&&\tfrac{1}{6}&&\tfrac{1}{3}&\\
\tfrac{1}{4}&&\tfrac{1}{12}&&\tfrac{1}{12}&&\tfrac{1}{4}\\
&&&\vdots&&&
\end{array}
\]
where $a_{n,1}=\frac{1}{n}$ and $a_{n,k+1}=a_{n-1,k}-a_{n,k}$. Find the remainder when the sum of the reciprocals of the $2007$ terms on the $2007^\text{th}$ row gets divided by $2008$.
Answer: 1944
Problem: For which $p$ prime numbers, there is an integer root of the polynomial $1 + p + Q(x^1) \cdot Q(x^2) \ldots Q(x^{2p - 2})$ such that $Q(x)$ is a polynomial with integer coefficients?
Answer: 2
Problem: Let $ABCD$ be a convex quadrilateral with $\angle DAB = \angle BDC = 90^\circ$. Let the incircles of triangles $ABD$ and $BCD$ touch $BD$ at $P$ and $Q$, respectively, with $P$ lying in between $B$ and $Q$. If $AD = 999$ and $PQ = 200$, then what is the sum of the radii of the incircles of triangles $ABD$ and $BDC$?
Answer: 799
Problem: Compute $\left|\sum_{i=1}^{2022} \sum_{j=1}^{2022} \cos\left(\frac{ij\pi}{2023}\right)\right|.$
Answer: 1011
Problem: Let $a_n$ be the least positive integer such that the sum of its digits is $n$. Find $a_1 + a_2 + a_3 + \dots + a_{20}$.
Answer: 1074
Problem: Let $2000 < N < 2100$ be an integer. Suppose the last day of year $N$ is a Tuesday while the first day of year $N+2$ is a Friday. The fourth Sunday of year $N+3$ is the $m$th day of January. What is $m$?
Answer: 23
Problem: Spencer is making burritos, each consisting of one wrap and one filling. He has enough filling for up to four beef burritos and three chicken burritos. However, he only has five wraps for the burritos. In how many ways can he make exactly five burritos?
Answer: 25
Problem: Suppose $n$ is an integer $\geq 2$. Determine the first digit after the decimal point in the decimal expansion of the number $\sqrt[3]{n^{3}+2n^{2}+n}$.
Answer: 6
Problem: Daniel writes a list of positive integers, each less than or equal to 10, on a board from top to bottom. Next to each number in Daniel's list, Martin writes the number of times that number appears in Daniel's list, creating a new list of the same length. When Martin's list is read from bottom to top, it matches the list Daniel wrote from top to bottom. Find the maximum possible length of Daniel's list.
Answer: 39
Problem: Let $L, E, T, M, \text{ and } O$ be digits that satisfy the equation $LEET + LMT = TOOL.$ Given that $O$ has the value of $0$, digits may be repeated, and $L \neq 0$, determine the value of the 4-digit integer $ELMO.$
Answer: 1880
Problem: Find the number of three-digit numbers such that each of the first two digits is divisible by the third digit.
Answer: 138
Problem: Let $ABC$ be a triangle with circumradius $17$, inradius $4$, circumcircle $\Gamma$, and $A$-excircle $\Omega$. Suppose the reflection of $\Omega$ over line $BC$ is internally tangent to $\Gamma$. Compute the area of $\triangle ABC$.
Answer: 128
Problem: $f(x)$ and $g(x)$ are linear functions such that for all $x$, $f(g(x)) = g(f(x)) = x$. If $f(0) = 4$ and $g(5) = 17$, compute $f(2006)$.
Answer: 122
Problem: Justine has two fair dice, one with sides labeled $1, 2, \ldots, m$ and one with sides labeled $1, 2, \ldots, n.$ She rolls both dice once. If $\frac{3}{20}$ is the probability that at least one of the numbers showing is at most 3, find the sum of all distinct possible values of $m+n$.
Answer: 996
Problem: Let $Q$ be a quadratic polynomial. If the sum of the roots of $Q^{100}(x)$ (where $Q^i(x)$ is defined by $Q^1(x)=Q(x)$, $Q^i(x)=Q(Q^{i-1}(x))$ for integers $i \geq 2$) is $8$ and the sum of the roots of $Q$ is $S$, compute $|\log_2(S)|$.
Answer: 96
Problem: Find the positive integer $n$ such that $32$ is the product of the real number solutions of the equation $x^{\log_2(x^3)-n} = 13$.
Answer: 15
Problem: Find the number of prime factors in the factorization of $12!$, where $12! = 12 \cdot 11 \cdot 10 \cdot 9 \cdot 8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1$.
Answer: 19
Problem: Eve randomly chooses two **distinct** points on the coordinate plane from the set of all $11^2$ lattice points $(x, y)$ with $0 \le x \le 10$, $0 \le y \le 10$. Then, Anne the ant walks from the point $(0,0)$ to the point $(10, 10)$ using a sequence of one-unit right steps and one-unit up steps. Let $P$ be the number of paths Anne could take that pass through both of the points that Eve chose. The expected value of $P$ is $\binom{20}{10} \cdot \frac{a}{b}$ for relatively prime positive integers $a$ and $b$. Compute $100a+b$.
Answer: 942
Problem: For how many positive integers $n \leq 100$ is it true that $10n$ has exactly three times as many positive divisors as $n$ has?
Answer: 28
Problem: Let $X = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$ and $A = \{1, 2, 3, 4\}$. Find the number of $4$-element subsets $Y$ of $X$ such that $10 \in Y$ and the intersection of $Y$ and $A$ is not empty.
Answer: 74
Problem: What are the last $3$ digits of $1! + 2! + \cdots + 100!$?
Answer: 313
Problem: Consider the $5 \times 5 \times 5$ equilateral triangular grid as shown:

```asy
size(5cm);
real n = 5;
for (int i = 0; i < n; ++i) {
    draw((0.5*i,0.866*i)--(n-0.5*i,0.866*i));
}
for (int i = 0; i < n; ++i) {
    draw((n-i,0)--((n-i)/2,(n-i)*0.866));
}
for (int i = 0; i < n; ++i) {
    draw((i,0)--((n+i)/2,(n-i)*0.866));
}
```

Ethan chooses two distinct upward-oriented equilateral triangles bounded by the gridlines. The probability that Ethan chooses two triangles that share exactly one vertex can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m+n$.
Answer: 23
Problem: Let $A_1, A_2, A_3, \ldots , A_{12}$ be the vertices of a regular $12$-gon (dodecagon). Find the number of points in the plane that are equidistant to at least $3$ distinct vertices of this $12$-gon.
Answer: 1
Problem: Let \( p(x) = c_1 + c_2 \cdot 2^x + c_3 \cdot 3^x + c_4 \cdot 5^x + c_5 \cdot 8^x \). Given that \( p(k) = k \) for \( k = 1, 2, 3, 4, 5 \), compute \( p(6) \).
Answer: -50
Problem: Let $BCB'C'$ be a rectangle, let $M$ be the midpoint of $B'C'$, and let $A$ be a point on the circumcircle of the rectangle. Let triangle $ABC$ have orthocenter $H$, and let $T$ be the foot of the perpendicular from $H$ to line $AM$. Suppose that $AM = 2$, $[ABC] = 2020$, and $BC = 10$. Then $AT = \frac{m}{n}$, where $m$ and $n$ are positive integers with $\gcd(m,n) = 1$. Compute $100m+n$.
Answer: 2102