ChenDRAG/VeRL_math_validation huggingface dataset:
Problem: Andy has 2010 square tiles, each with a side length of one unit. He plans to arrange the tiles in an $m \times n$ rectangle, where $mn = 2010$. Compute the sum of the perimeters of all the different possible rectangles he can make. Two rectangles are considered the same if one can be rotated to become the other. For example, a $1 \times 2010$ rectangle is considered the same as a $2010 \times 1$ rectangle.
Answer: 9792
Problem: How many divisors of $30^{2003}$ are there which do not divide $20^{2000}$?
Answer: 8044086060
Problem: We color all the sides and diagonals of a regular polygon $P$ with 43 vertices either red or blue. Each vertex is an endpoint of 20 red segments and 22 blue segments. A triangle formed by vertices of $P$ is called monochromatic if all of its sides have the same color. Suppose there are 2022 blue monochromatic triangles. How many red monochromatic triangles are there?
Answer: 859
Problem: In triangle $ABC$, where $AB = 13$, $BC = 14$, and $CA = 15$, the segment $BC$ is divided into $n+1$ congruent segments by $n$ points. Among these points are the feet of the altitude, median, and angle bisector from vertex $A$. Determine the smallest possible value of $n$.
Answer: 27
Problem: Let $x_1$, $x_2$, \ldots, $x_{10}$ be 10 numbers. Suppose that $x_i + 2x_{i+1} = 1$ for each $i$ from 1 through 9. What is the value of $x_1 + 512x_{10}$?
Answer: 171
Problem: What is the greatest number of positive integers less than or equal to 2016 that can be chosen such that no two of them differ by 1, 2, or 6?
Answer: 576
Problem: How many ways are there to express $1,000,000$ as a product of exactly three integers greater than $1$? For the purpose of this problem, $abc$ is not considered different from $bac$, etc.
Answer: 114
Problem: Compute $ax^{2018} + by^{2018}$, given that there exist real numbers $a$, $b$, $x$, and $y$ which satisfy the following four equations:
\[
\begin{align*}
ax^{2014} + by^{2014} &= 6, \\
ax^{2015} + by^{2015} &= 7, \\
ax^{2016} + by^{2016} &= 3, \\
ax^{2017} + by^{2017} &= 50.
\end{align*}
\]
Answer: -417
Problem: Six lines are drawn in the plane. Determine the maximum number of points through which at least three lines pass.
Answer: 4
Problem: In how many rearrangements of the numbers $1, 2, 3, 4, 5, 6, 7, 8, 9$ do the numbers form a \textit{hill}? A \textit{hill} is defined as a sequence where the numbers increase to a peak and then decrease, such as in $129876543$ or $258976431$.
Answer: 254
Problem: Given that the base-17 integer $\overline{8323a02421_{17}}$ (where $a$ is a base-17 digit) is divisible by $16_{10}$, find $a$. Express your answer in base $10$.
Answer: 7
Problem: In triangle $ABC$, points $D$, $E$, and $F$ are on sides $BC$, $CA$, and $AB$ respectively, such that $BF = BD = CD = CE = 5$ and $AE - AF = 3$. Let $I$ be the incenter of $\triangle ABC$. The circumcircles of $\triangle BFI$ and $\triangle CEI$ intersect at $X \neq I$. Find the length of $DX$.
Answer: 3
Problem: Trapezoid $ABCD$ is an isosceles trapezoid with $AD = BC$. Point $P$ is the intersection of the diagonals $AC$ and $BD$. If the area of $\triangle ABP$ is $50$ and the area of $\triangle CDP$ is $72$, what is the area of the entire trapezoid?
Answer: 242
Problem: Let $\Gamma_1$, $\Gamma_2$ be two circles, where $\Gamma_1$ has a smaller radius, intersecting at two points $A$ and $B$. Points $C$ and $D$ lie on $\Gamma_1$ and $\Gamma_2$ respectively such that point $A$ is the midpoint of the segment $CD$. Line $CB$ intersects the circle $\Gamma_2$ for the second time at point $F$, and line $DB$ intersects the circle $\Gamma_1$ for the second time at point $E$. The perpendicular bisectors of segments $CD$ and $EF$ intersect at point $P$. Knowing that $CA = 12$ and $PE = 5$, find $AP$.
Answer: 13
Problem: Triangle $ABC$ has $\overline{AB} = 5$, $\overline{BC} = 4$, $\overline{CA} = 6$. Points $D$ and $E$ are on sides $AB$ and $AC$, respectively, such that $\overline{AD} = \overline{AE} = \overline{BC}$. Let $CD$ and $BE$ intersect at $F$, and let $AF$ and $DE$ intersect at $G$. The length of $FG$ can be expressed in the form $\frac{a\sqrt{b}}{c}$ in simplified form. What is $a + b + c$?
Answer: 255
Problem: Let $ABCD$ be a unit square. Points $M$ and $N$ are the midpoints of sides $AB$ and $BC$ respectively. Let $P$ and $Q$ be the midpoints of line segments $AM$ and $BN$ respectively. Find the reciprocal of the area of the triangle enclosed by the three line segments $PQ$, $MN$, and $DB$.
Answer: 128
Problem: For positive integers $m$ and $n$, the decimal representation for the fraction $\frac{m}{n}$ begins with $0.711$ followed by other digits. Find the least possible value for $n$.
Answer: 45
Problem: Greta is completing an art project. She has twelve sheets of paper: four red, four white, and four blue. She also has twelve paper stars: four red, four white, and four blue. She randomly places one star on each sheet of paper. The probability that no star will be placed on a sheet of paper that is the same color as the star is $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $n - 100m.$
Answer: 25
Problem: Compute the sum:
\[ S = 1 + 2 + 3 - 4 - 5 + 6 + 7 + 8 - 9 - 10 + \dots - 2010 \]
where every three consecutive \( + \) are followed by two \( - \).
Answer: 401799
Problem: If we divide $2020$ by a prime $p$, the remainder is $6$. Determine the largest possible value of $p$.
Answer: 53
Problem: Let $u=\{a_1, \ldots, a_5\}$, where all $a_i \in \{0,1\}$. Let $D(u)$ be the set of all subsequences of $u$ with length 4. Let $A$ be a set of sequences $u_1, \ldots, u_{|A|}$ with the property that $|D(u_i) \cap D(u_j)| = 0$ for all $i \neq j$. What is the maximum value of $|A|$?
Answer: 5
Problem: Let $ABC$ be a triangle with incenter $I$ such that $AB = 20$ and $AC = 19$. Point $P \neq A$ lies on line $AB$ and point $Q \neq A$ lies on line $AC$. Suppose that $IA = IP = IQ$ and that line $PQ$ passes through the midpoint of side $BC$. Suppose that $BC = \frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $100m + n$.
Answer: 3902
Problem: A computer outputs the values of the expression $(n+1) \cdot 2^n$ for $n = 1, n = 2, n = 3$, etc. What is the largest number of consecutive values that are perfect squares?
Answer: 2
Problem: For any positive integer $n$, define 
\[ E(n) = n(n+1)(2n+1)(3n+1)\cdots (10n+1). \]
Find the greatest common divisor of $E(1), E(2), E(3), \dots, E(2009)$.
Answer: 2310
Problem: The number $734,851,474,594,578,436,096$ is equal to $n^6$ for some positive integer $n$. What is the value of $n$?
Answer: 3004
Problem: Find the unique 3-digit number $N=\underline{A}\ \underline{B}\ \underline{C}$, whose digits $(A, B, C)$ are all nonzero, with the property that the product $P=\underline{A}\ \underline{B}\ \underline{C} \times \underline{A}\ \underline{B} \times \underline{A}$ is divisible by $1000$.
Answer: 875
Problem: Giraldo wrote five distinct natural numbers on the vertices of a pentagon. Next, he wrote on each side of the pentagon the least common multiple (LCM) of the numbers written on the two vertices of that side. He noticed that the five numbers written on the sides were equal. What is the smallest number Giraldo could have written on the sides?
Answer: 30
Problem: Paula rolls three standard fair dice. The probability that the three numbers rolled on the dice are the side lengths of a triangle with positive area is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
Answer: 109
Problem: Triangle $ABC$ has side lengths $AB = 5$, $BC = 12$, and $CA = 13$. Circle $\Gamma$ is centered at point $X$ exterior to triangle $ABC$ and passes through points $A$ and $C$. Let $D$ be the second intersection of $\Gamma$ with segment $\overline{BC}$. If $\angle BDA = \angle CAB$, the radius of $\Gamma$ can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m+n$.
Answer: 193
Problem: There are 25 ants on a number line; five at each of the coordinates $1$, $2$, $3$, $4$, and $5$. Each minute, one ant moves from its current position to a position one unit away. What is the minimum number of minutes that must pass before it is possible for no two ants to be on the same coordinate?
Answer: 126
Problem: Right triangle $\triangle ABC$ with a right angle at $A$ has $AB = 20$ and $AC = 15$. Point $D$ is on $AB$ with $BD = 2$. Points $E$ and $F$ are placed on ray $CA$ and ray $CB$, respectively, such that $CD$ is a median of $\triangle CEF$. Find the area of $\triangle CEF$.
Answer: 54
Problem: Find the smallest positive integer that is relatively prime to each of $2$, $20$, $204$, and $2048$.
Answer: 1
Problem: We say that a $2023$-tuple of nonnegative integers $(a_1,\ldots,a_{2023})$ is \textit{sweet} if the following conditions hold:

- $a_1+a_2+\ldots+a_{2023}=2023$
- $\frac{a_1}{2}+\frac{a_2}{2^2}+\ldots+\frac{a_{2023}}{2^{2023}}\le 1$

Determine the greatest positive integer $L$ so that $a_1+2a_2+\ldots+2023a_{2023}\ge L$ holds for every sweet $2023$-tuple $(a_1,\ldots,a_{2023})$.
Answer: 22228
Problem: Find the 3-digit positive integer that has the most divisors.
Answer: 840
Problem: Compute the number of ordered quintuples of nonnegative integers \((a_1, a_2, a_3, a_4, a_5)\) such that \(0 \leq a_1, a_2, a_3, a_4, a_5 \leq 7\) and \(5\) divides \(2^{a_1} + 2^{a_2} + 2^{a_3} + 2^{a_4} + 2^{a_5}\).
Answer: 6528
Problem: Consider the numbers $\{24, 27, 55, 64, x\}$. Given that the mean of these five numbers is prime and the median is a multiple of $3$, compute the sum of all possible positive integral values of $x$.
Answer: 60
Problem: A moth starts at vertex $A$ of a certain cube and is trying to get to vertex $B$, which is opposite $A$, in five or fewer "steps," where a step consists of traveling along an edge from one vertex to another. The moth will stop as soon as it reaches $B$. How many ways can the moth achieve its objective?
Answer: 48
Problem: Define a *small* prime to be a prime under $1$ billion. Find the sum of all *small* primes of the form $20^n + 1$, given that the answer is greater than $1000$.
Answer: 160402
Problem: Using each of the eight digits $1, 3, 4, 5, 6, 7, 8,$ and $9$ exactly once, form a three-digit number $A$, two two-digit numbers $B$ and $C$ (where $B < C$), and a one-digit number $D$. The numbers are such that $A + D = B + C = 143$. In how many ways can this be done?
Answer: 24
Problem: Let $n$ be the largest integer that is the product of exactly 3 distinct prime numbers: $x$, $y$, and $10x+y$, where $x$ and $y$ are digits. What is the sum of the digits of $n$?
Answer: 12
Problem: A regular $2015$-simplex $\mathcal{P}$ has $2016$ vertices in $2015$-dimensional space such that the distances between every pair of vertices are equal. Let $S$ be the set of points contained inside $\mathcal{P}$ that are closer to its center than any of its vertices. The ratio of the volume of $S$ to the volume of $\mathcal{P}$ is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find the remainder when $m+n$ is divided by $1000$.
Answer: 321
Problem: Let $M$ be the maximum possible value of $x_1x_2 + x_2x_3 + \cdots + x_5x_1$ where $x_1, x_2, \cdots, x_5$ is a permutation of $(1, 2, 3, 4, 5)$. Let $N$ be the number of permutations for which this maximum is attained. Evaluate $M + N$.
Answer: 58
Problem: $ABCD$ is a unit square. If $\angle PAC = \angle PCD$, find the length $BP$.
Answer: 1
Problem: The Tournament of Towns is held once per year. This time, the year of its autumn round is divisible by the number of the tournament: $2021 \div 43 = 47$. How many more times will humanity witness such a wonderful event?
Answer: 4
Problem: For an integer $30 \le k \le 70$, let $M$ be the maximum possible value of:
\[ \frac{A}{\gcd(A,B)} \]
where $A = \binom{100}{k}$ and $B = \binom{100}{k+3}$. Find the remainder when $M$ is divided by $1000$.
Answer: 664
Problem: Suppose that $(\overline{AB}, \overline{CD})$ is a pair of two-digit positive integers, where the digits $A$ and $C$ must be nonzero. The product $\overline{AB} \cdot \overline{CD}$ divides the four-digit number $\overline{ABCD}$. Find the sum of all possible values of the three-digit number $\overline{ABC}$.
Answer: 308
Problem: It is known that $2018(2019^{39} + 2019^{37} + \cdots + 2019) + 1$ is prime. How many positive factors does $2019^{41} + 1$ have?
Answer: 24
Problem: Let $S$ be the sum of all $x$ such that $1 \leq x \leq 99$ and \[ \{x^2\} = \{x\}^2. \] Compute $\lfloor S \rfloor$.
Answer: 641999
Problem: Rectangle $ABCD$ has $AB = 8$ and $BC = 13$. Points $P_1$ and $P_2$ lie on $AB$ and $CD$ with $P_1P_2 \parallel BC$. Points $Q_1$ and $Q_2$ lie on $BC$ and $DA$ with $Q_1Q_2 \parallel AB$. Find the area of quadrilateral $P_1Q_1P_2Q_2$.
Answer: 52
Problem: Find the smallest three-digit number such that the following holds: If the order of the digits of this number is reversed, and the number obtained is added to the original number, the resulting number consists of only odd digits.
Answer: 209
Problem: Alexa wrote the first $16$ numbers of a sequence:
\[1, 2, 2, 3, 4, 4, 5, 6, 6, 7, 8, 8, 9, 10, 10, 11, \ldots\]
Then she continued following the same pattern, until she had $2015$ numbers in total. 
What was the last number she wrote?
Answer: 1344
Problem: Convex quadrilateral $ABCD$ has perpendicular diagonals and satisfies the conditions $\angle B = \angle C = 90^\circ$, $BC = 20$, and $AD = 30$. Compute the square of the area of a triangle with side lengths equal to $CD$, $DA$, and $AB$.
Answer: 30000
Problem: Find the largest integer $n$ which equals the product of its leading digit and the sum of its digits.
Answer: 48
Problem: Let $m$ be the maximum possible value of $x^{16} + \frac{1}{x^{16}}$, where \[x^6 - 4x^4 - 6x^3 - 4x^2 + 1=0.\] Find the remainder when $m$ is divided by $2007$.
Answer: 1865
Problem: Triangle $LMT$ has $\overline{MA}$ as an altitude. Given that $MA = 16$, $MT = 20$, and $LT = 25$, find the length of the altitude from $L$ to $\overline{MT}$.
Answer: 20
Problem: Determine the value of $f_{1991}(2^{1990})$ where for a given positive integer $k$, the function $f_1(k)$ is defined as the square of the sum of its digits. The sequence is recursively defined as $f_{n+1}(k) = f_1(f_n(k))$.
Answer: 256
Problem: The diagram shows a large circular dartboard with four smaller shaded circles, each internally tangent to the larger circle. Two of the internal circles have half the radius of the large circle and are tangent to each other. The other two smaller circles are tangent to these circles. If a dart is thrown such that it sticks to a point randomly chosen on the dartboard, then the probability that the dart sticks to a point in the shaded area is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
Answer: 31
Problem: Find the number of ordered pairs of positive integers $(a,b)$ such that $a+b$ is a prime number, $1 \leq a, b \leq 100$, and $\frac{ab+1}{a+b}$ is an integer.
Answer: 51
Problem: Find the number of ways to permute the elements of the set $\{1,2,3,4,5,6,7,8,9\}$ such that no two adjacent numbers are both even or both odd.
Answer: 2880
Problem: Unconventional dice are to be designed such that the six faces are marked with numbers from $1$ to $6$, with $1$ and $2$ appearing on opposite faces. Furthermore, each face is colored either red or yellow, with opposite faces always of the same color. Two dice are considered to have the same design if one of them can be rotated to obtain a dice that has the same numbers and colors on the corresponding faces as the other one. Find the number of distinct dice that can be designed.
Answer: 48
Problem: For a positive integer $n$, define the $n$th triangular number $T_n$ to be $\frac{n(n+1)}{2}$, and define the $n$th square number $S_n$ to be $n^2$. Find the value of 
\[ \sqrt{S_{62}+T_{63}\sqrt{S_{61}+T_{62}\sqrt{\cdots \sqrt{S_2+T_3\sqrt{S_1+T_2}}}}}. \]
Answer: 1954
Problem: The area of the triangle whose altitudes have lengths $36.4$, $39$, and $42$ can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
Answer: 3553
Problem: For a positive integer $n$, let $f(n)$ be the number of (not necessarily distinct) primes in the prime factorization of $k$. For example, $f(1) = 0$, $f(2) = 1$, and $f(4) = f(6) = 2$. Let $g(n)$ be the number of positive integers $k \leq n$ such that $f(k) \geq f(j)$ for all $j \leq n$. Find $g(1) + g(2) + \ldots + g(100)$.
Answer: 136
Problem: A positive integer $n$ is called a *primary divisor* if for every positive divisor $d$ of $n$, at least one of the numbers $d - 1$ and $d + 1$ is prime. For example, $8$ is a primary divisor because its positive divisors $1$, $2$, $4$, and $8$ each differ by $1$ from a prime number ($2$, $3$, $5$, and $7$, respectively), while $9$ is not a primary divisor because the divisor $9$ does not differ by $1$ from a prime number (both $8$ and $10$ are composite). Determine the largest primary divisor number.
Answer: 96
Problem: Find all prime numbers $p$ for which $3^p - (p+2)^2$ is also prime.
Answer: 3
Problem: The number $2.5081081081081 \ldots$ can be written as $\frac{m}{n}$ where $m$ and $n$ are natural numbers with no common factors. Find $m + n$.
Answer: 649
Problem: Consider an odd prime $p$. For each $i$ in the set \( \{1, 2, \ldots, p-1\} \), let \( r_i \) be the remainder of \( i^p \) when it is divided by \( p^2 \). Find the sum:
\[
r_1 + r_2 + \ldots + r_{p-1}
\]
Answer: 0
Problem: Find the sum of all positive integers $n$ such that $1 + 2 + \cdots + n$ divides 
\[
15 \left[ (n+1)^2 + (n+2)^2 + \cdots + (2n)^2 \right].
\]
Answer: 64
Problem: In a 14-team baseball league, each team played every other team 10 times. At the end of the season, the number of games won by each team differed from those won by the team immediately following it by the same amount. Determine the greatest number of games the last-place team could have won, assuming that no ties were allowed.
Answer: 52
Problem: Find the number of integers $n > 1$ which divide $a^{25} - a$ for every integer $a$.
Answer: 31
Problem: 30 students participated in the mathematical Olympiad. Each of them was given 8 problems to solve. The jury estimated their work with the following rule:
1) Each problem was worth \( k \) points, if it wasn't solved by exactly \( k \) students;
2) Each student received the maximum possible points in each problem or got 0 in it.

Lasha got the least number of points. What's the maximal number of points he could have?

**Remark:** Rule 1 means that if the problem was solved by exactly \( k \) students, then each of them got \( 30 - k \) points in it.
Answer: 60
Problem: Consider the points $A_1$ and $A_2$ on the side $AB$ of the square $ABCD$ such that $|AB| = 3 |AA_1|$ and $|AB| = 4 |A_2B|$. Similarly, consider points $B_1$ and $B_2$ on $BC$, $C_1$ and $C_2$ on $CD$, and $D_1$ and $D_2$ on $DA$.

The intersection point of lines $D_2A_1$ and $A_2B_1$ is $E$.
The intersection point of lines $A_2B_1$ and $B_2C_1$ is $F$.
The intersection point of lines $B_2C_1$ and $C_2D_1$ is $G$.
The intersection point of lines $C_2D_1$ and $D_2A_1$ is $H$.

Find the area of the square $EFGH$, given that the area of $ABCD$ is $1$.
Answer: 1
Problem: Consider a board with $n$ rows and $4$ columns. In the first line, there are $4$ zeros (one in each column). Each subsequent line is obtained from the previous line by performing the following operation: one of the columns, which you can choose, is maintained as in the previous line; the other three are changed:
- If in the previous line there was a $0$, then in the next line, a $1$ is placed.
- If in the previous line there was a $1$, then in the next line, a $2$ is placed.
- If in the previous line there was a $2$, then in the next line, a $0$ is placed.

Build the largest possible board with all its lines distinct and demonstrate that it is impossible to build a larger board.
Answer: 27
Problem: Find the number of ordered triples of positive integers $(a, b, c)$, where $a, b, c$ form a strictly increasing arithmetic progression, $a + b + c = 2019$, and there exists a triangle with side lengths $a, b$, and $c$.
Answer: 336
Problem: Let $x_1 = \frac{1}{20}$, $x_2 = \frac{1}{13}$, and \[ x_{n+2} = \frac{2x_nx_{n+1}(x_n + x_{n+1})}{x_n^2 + x_{n+1}^2} \] for all integers $n \geq 1$. Evaluate $\sum_{n=1}^\infty \left( \frac{1}{x_n + x_{n+1}} \right)$.
Answer: 23
Problem: Let $Q$ be a polynomial \[Q(x)=a_0+a_1x+\cdots+a_nx^n,\] where $a_0,\ldots,a_n$ are nonnegative integers. Given that $Q(1)=4$ and $Q(5)=152$, find $Q(6)$.
Answer: 254
Problem: Hen Hao randomly selects two distinct squares on a standard $8 \times 8$ chessboard. Given that the two squares touch (at either a vertex or a side), the probability that the two squares are the same color can be expressed in the form $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Find $100m + n$.
Answer: 715
Problem: We are given some three-element subsets of $\{1,2, \dots ,n\}$ for which any two of them have at most one common element. We call a subset of $\{1,2, \dots ,n\}$ nice if it doesn't include any of the given subsets. If, no matter how the three-element subsets are selected initially, we can add one more element to every 29-element nice subset while keeping it nice, find the minimum value of $n$.
Answer: 436
Problem: In the diagram below, $ABCDE$ is a regular pentagon. The line $\overline{AG}$ is perpendicular to $\overline{CD}$, and $\overline{BD}$ intersects $\overline{AG}$ at point $F$. Find the degree measure of $\angle AFB$. 

\begin{asy}
import math;
size(4cm);
pen dps = fontsize(10);
defaultpen(dps);
pair A,B,C,D,E,F,G;
A=dir(90);
B=dir(162);
C=dir(234);
D=dir(306);
E=dir(18);
F=extension(A,G,B,D);
G=(C+D)/2;
draw(A--B--C--D--E--cycle^^A--G^^B--D);
label("$A$",A,dir(90)*0.5);
label("$B$",B,dir(162)*0.5);
label("$C$",C,dir(234)*0.5);
label("$D$",D,dir(306)*0.5);
label("$E$",E,dir(18)*0.5);
label("$F$",F,NE*0.5);
label("$G$",G,S*0.5);
\end{asy}
Answer: 54
Problem: Find the sum of the values of $x$ for which $\binom{x}{0} - \binom{x}{1} + \binom{x}{2} - \cdots + \binom{x}{2008} = 0$.
Answer: 2017036
Problem: How many ways are there to arrange the numbers $1$ through $8$ into a $2 \times 4$ grid such that:
1. The sum of the numbers in each of the two rows is a multiple of $6$.
2. The sum of the numbers in each of the four columns is a multiple of $3$?
Answer: 288
Problem: The largest term in the binomial expansion of $(1 + \frac{1}{2})^{31}$ is of the form $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers. What is the value of $b$? 

As an example of a binomial expansion, the binomial expansion of an expression of the form $(x + y)^3$ is the sum of four terms: 
\[ x^3 + 3x^2y + 3xy^2 + y^3. \]
Answer: 1024
Problem: Find all 3-digit numbers \( \overline{abc} \) such that \( \overline{abc} = a! + b! + c! \).
Answer: 145
Problem: Let $f : \mathbb{N} \to \mathbb{N}$ be a function such that $f(f(1995)) = 95$, $f(xy) = f(x)f(y)$, and $f(x) \le x$ for all $x, y$. Find all possible values of $f(1995)$.
Answer: 95
Problem: Sitting at a desk, Alice writes a nonnegative integer $N$ on a piece of paper, with $N \le 10^{10}$. Interestingly, Celia, sitting opposite Alice at the desk, is able to properly read the number upside-down and gets the same number $N$, without any leading zeros. (Note that the digits 2, 3, 4, 5, and 7 will not be read properly when turned upside-down.) Find the number of possible values of $N$.
Answer: 4999
Problem: A circle with a radius of $320$ is tangent to the inside of a circle with a radius of $1000$. The smaller circle is tangent to a diameter of the larger circle at a point $P$. How far is the point $P$ from the outside of the larger circle?
Answer: 400
Problem: Determine the smallest possible value of
$$|2^m - 181^n|,$$
where $m$ and $n$ are positive integers.
Answer: 7
Problem: Let $p_1, p_2, p_3, p_4$ be four distinct prime numbers, and let $1 = d_1 < d_2 < \ldots < d_{16} = n$ be the divisors of $n = p_1 p_2 p_3 p_4$. Determine all $n < 2001$ such that $d_9 - d_8 = 22$.
Answer: 1995
Problem: Let $s(n)$ denote the number of $1$'s in the binary representation of $n$. Compute \[ \frac{1}{255}\sum_{0\leq n<16}2^n(-1)^{s(n)}. \]
Answer: 45
Problem: Compute $99(99^2+3) + 3 \cdot 99^2$.
Answer: 999999