sungyub/dapo-math-17k-verl huggingface dataset:
user: 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
user: $ABCD$ is a unit square. If $\angle PAC = \angle PCD$, find the length $BP$.

Answer: 1
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: Find the largest integer $n$ which equals the product of its leading digit and the sum of its digits.

Answer: 48
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: Find all prime numbers $p$ for which $3^p - (p+2)^2$ is also prime.

Answer: 3
user: 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
user: 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
user: 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
user: 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
user: Find the number of integers $n > 1$ which divide $a^{25} - a$ for every integer $a$.

Answer: 31
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: How many ways are there to arrange the numbers $1$ through $8$ into a $2 \times 4$ grid such that:
The sum of the numbers in each of the two rows is a multiple of $6$.
The sum of the numbers in each of the four columns is a multiple of $3$?

Answer: 288
user: 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
user: Find all 3-digit numbers \( \overline{abc} \) such that \( \overline{abc} = a! + b! + c! \).

Answer: 145
user: 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
user: 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
user: 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
user: Determine the smallest possible value of
$$|2^m - 181^n|,$$
where $m$ and $n$ are positive integers.

Answer: 7
user: 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
user: 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
user: Compute $99(99^2+3) + 3 \cdot 99^2$.

Answer: 999999
user: Find the least positive integer which is a multiple of $13$ and all its digits are the same.

Answer: 111111
user: For a positive real number $x$, let $[x]$ denote its integer part. For example, $[2.711] = 2$, $[7] = 7$, $[6.9] = 6$. 

Find the maximum real number $z$ such that $\left[\frac{5}{z}\right] + \left[\frac{6}{z}\right] = 7$. 

Calculate the value of $20z$.

Answer: 30
user: A triangle $ABC$ is given where $\angle BAC = 40^\circ$ and $\angle ABC = 20^\circ$. Find the length of the angle bisector drawn from vertex $C$, given that the sides $AB$ and $BC$ differ by $4$ centimeters.

Answer: 4
user: Using the digits $1, 2, 3, 4, 5, 6$ without repetition, form three two-digit numbers. Calculate the number of different sums that can be obtained when these numbers are added together.

Answer: 10
user: Let $f(n)$ be the number of times you have to hit the $\sqrt{}$ key on a calculator to get a number less than $2$ starting from $n$. For instance, $f(2) = 1$, $f(5) = 2$. For how many $1 < m < 2008$ is $f(m)$ odd?

Answer: 242
user: Find the number of distinct values obtained from the expression:

$$ \frac{n^2 - 2}{n^2 - n + 2} $$

where \( n \in \{1, 2, 3, \ldots, 100\} \).

Answer: 98
user: In the following alpha-numeric puzzle, each letter represents a different non-zero digit. What are all possible values for $b+e+h$?

$$ \begin{array}{cccc}
 & a & b & c \\
 & d & e & f \\
+ & g & h & i \\
\hline
1 & 6 & 6 & 5 \\
\end{array} $$

Answer: 15
user: The figure below was created by joining together 5 non-overlapping congruent squares. The total area of the figure is 45. Determine the perimeter of the figure.

Answer: 36
user: Compute the sum of all positive integers $a \leq 26$ for which there exist integers $b$ and $c$ such that $a + 23b + 15c - 2$ and $2a + 5b + 14c - 8$ are both multiples of $26$.

Answer: 31
user: Let $P(x) = x^3 + 8x^2 - x + 3$ and let the roots of $P$ be $a, b,$ and $c.$ The roots of a monic polynomial $Q(x)$ are $ab - c^2, ac - b^2, bc - a^2.$ Find $Q(-1).$

Answer: 1536
user: If 
$$\sum_{k=1}^{40} \left( \sqrt{1 + \frac{1}{k^{2}} + \frac{1}{(k + 1)^{2}}}\right) = a + \frac {b}{c}$$
where $a, b, c \in \mathbb{N}, b < c, \gcd(b,c) = 1,$ then what is the value of $a + b?$

Answer: 80
user: Let $x_0, x_1, \ldots$ be a sequence of real numbers such that \(x_n = \frac{1 + x_{n -1}}{x_{n - 2}}\) for \(n \geq 2\). Find the number of ordered pairs of positive integers \((x_0, x_1)\) such that the sequence gives \(x_{2018} = \frac{1}{1000}\).

Answer: 128
user: Today is Barbara's birthday, and Alberto wants to give her a gift by playing the following game. The numbers $0, 1, 2, \ldots, 1024$ are written on a blackboard. 

First, Barbara erases $2^{9}$ numbers.
Then, Alberto erases $2^{8}$ numbers.
Next, Barbara erases $2^{7}$ numbers.
This process continues until there are only two numbers $a$ and $b$ left.

Barbara earns $|a-b|$ euros. 

Find the maximum number of euros that Barbara can always win, regardless of Alberto's strategy.

Answer: 32
user: Let $P(x)$ be the unique polynomial of degree four for which $P(165) = 20$, and \[ P(42) = P(69) = P(96) = P(123) = 13. \] Compute $P(1) - P(2) + P(3) - P(4) + \dots + P(165)$.

Answer: 20
user: Natural numbers $a \leq b \leq c \leq d$ satisfy $a + b + c + d = 30$. Find the maximum value of the product $P = abcd$.

Answer: 3136
user: For positive real numbers $x$ and $y$, define their special mean to be the average of their arithmetic and geometric means. Find the total number of pairs of integers $(x, y)$, with $x \le y$, from the set of numbers $\{1,2,\ldots,2016\}$, such that the special mean of $x$ and $y$ is a perfect square.

Answer: 506
user: Determine the maximal size of a set of positive integers with the following properties:

The integers consist of digits from the set $\{ 1,2,3,4,5,6\}$.

No digit occurs more than once in the same integer.

The digits in each integer are in increasing order.

Any two integers have at least one digit in common (possibly at different positions).

There is no digit which appears in all the integers.

Answer: 32
user: The polynomial $f(x) = x^3 + rx^2 + sx + t$ has $r$, $s$, and $t$ as its roots (with multiplicity), where $f(1)$ is rational and $t \neq 0$. Compute $|f(0)|$.

Answer: 1
user: Positive integers $a$ and $b$ satisfy the equations $a^3 + 32b + 2c = 2018$ and $b^3 + 32a + 2c = 1115$. Find $a^2 + b^2 + c^2$.

Answer: 226
user: On a blackboard, a stranger writes the values of $s_7(n)^2$ for $n=0,1,\ldots,7^{20}-1$, where $s_7(n)$ denotes the sum of digits of $n$ in base $7$. Compute the average value of all the numbers on the board.

Answer: 3680
user: Let $a, b, c, x, y,$ and $z$ be complex numbers such that:
\[ a = \frac{b+c}{x-2}, \quad b = \frac{c+a}{y-2}, \quad c = \frac{a+b}{z-2}. \]
Given that:
\[ xy + yz + zx = 1000 \quad \text{and} \quad x + y + z = 2016, \]
find the value of $xyz$.

Answer: -4044
user: On a board, there are integers written from $1$ to $119$. Two players, $A$ and $B$, take turns making a move. A move consists of erasing $9$ numbers from the board. The player after whose move two numbers remain on the board wins, and their score is equal to the positive difference between the two remaining numbers. Player $A$ makes the first move. Find the highest integer $k$ such that player $A$ can ensure that their score is not smaller than $k$.

Answer: 64
user: Five rays $\overrightarrow{OA}$, $\overrightarrow{OB}$, $\overrightarrow{OC}$, $\overrightarrow{OD}$, and $\overrightarrow{OE}$ radiate in a clockwise order from $O$ forming four non-overlapping angles such that:
- $\angle EOD = 2\angle COB$,
- $\angle COB = 2\angle BOA$,
- $\angle DOC = 3\angle BOA$.

If $E$, $O$, $A$ are collinear with $O$ between $A$ and $E$, what is the degree measure of $\angle DOB$?

Answer: 90
user: Andrew rolls two fair six-sided dice, each numbered from $1$ to $6$. Brian rolls one fair twelve-sided die, numbered from $1$ to $12$. The probability that the sum of the numbers obtained from Andrew's two rolls is less than the number obtained from Brian's roll can be expressed as a common fraction in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

Answer: 17
user: In triangle $ABC$, where $AB = 52$, $BC = 56$, and $CA = 60$, let $D$ be the foot of the altitude from $A$, and $E$ be the intersection of the internal angle bisector of $\angle BAC$ with $BC$. Find the length of $DE$.

Answer: 6
user: PUMaCDonalds, a newly-opened fast food restaurant, has 5 menu items. If the first 4 customers each choose one menu item at random, what is the probability that the 4th customer orders a previously unordered item? The probability is given by \( \frac{m}{n} \), where \( m \) and \( n \) are relatively prime positive integers. Find \( m+n \).

Answer: 189
user: We call a path Valid if:

It only comprises the following types of steps:
 - $(x, y) \rightarrow (x + 1, y + 1)$
 - $(x, y) \rightarrow (x + 1, y - 1)$

It never goes below the x-axis.

Let $M(n)$ be the set of all valid paths from $(0,0)$ to $(2n,0)$, where $n$ is a natural number. 

Consider a Valid path $T \in M(n)$. Denote:

$$\phi(T) = \prod_{i=1}^{2n} \mu_i$$

where $\mu_i$ is defined as:

- $1$, if the $i^{th}$ step is $(x, y) \rightarrow (x + 1, y + 1)$
- $y$, if the $i^{th}$ step is $(x, y) \rightarrow (x + 1, y - 1)$

Now, let $f(n) = \sum_{T \in M(n)} \phi(T)$. Evaluate the number of zeroes at the end of the decimal expansion of $f(2021)$.

Answer: 0
user: David, Kevin, and Michael each choose an integer from the set $\{1, 2, \ldots, 100\}$ randomly, uniformly, and independently of each other. The probability that the positive difference between David's and Kevin's numbers is \emph{strictly} less than that of Kevin's and Michael's numbers is $\frac{m}{n}$, for coprime positive integers $m$ and $n$. Find $100m + n$.

Answer: 1005100
user: A substring of a number $n$ is a number formed by removing some digits from the beginning and end of $n$ (possibly a different number of digits is removed from each side). Find the sum of all prime numbers $p$ such that any substring of $p$ is also prime.

Answer: 576
user: Three roots of the quartic polynomial $f(x)=x^4 + ax^3 + bx + c$ are $-1$, $3$, and $5$. What is $a + b - c$?

Answer: 1
user: Mr. Ambulando is at the intersection of $5^{\text{th}}$ and $\text{A St}$, and needs to walk to the intersection of $1^{\text{st}}$ and $\text{F St}$. There's an accident at the intersection of $4^{\text{th}}$ and $\text{B St}$, which he'd like to avoid. 

Given that Mr. Ambulando wants to walk the shortest distance possible, how many different routes through downtown can he take?

Answer: 56
user: Find the largest positive integer $k$ such that $\phi(\sigma(2^k)) = 2^k$. Here, $\phi(n)$ denotes the number of positive integers that are smaller than $n$ and relatively prime to $n$, and $\sigma(n)$ denotes the sum of divisors of $n$. You are given the hint that $641 \mid 2^{32} + 1$.

Answer: 31
user: There exists a unique pair of polynomials $P(x)$ and $Q(x)$ such that:
\[ \begin{align*} P(Q(x)) &= P(x)(x^2-6x+7) \\ Q(P(x)) &= Q(x)(x^2-3x-2) \end{align*} \]
Compute $P(10) + Q(-10)$.

Answer: -90
user: Let $S$ be the set of natural numbers whose digits are different and belong to the set $\{1, 3, 5, 7\}$. Calculate the sum of the elements of $S$.

Answer: 117856
user: How many ways are there to choose three digits $A, B, C$ with $1 \leq A \leq 9$ and $0 \leq B, C \leq 9$ such that $\overline{ABC}_b$ is even for all choices of base $b$ with $b \geq 10$?

Answer: 225
user: Let $X_0$ be the interior of a triangle with side lengths $3$, $4$, and $5$. For all positive integers $n$, define $X_n$ to be the set of points within $1$ unit of some point in $X_{n-1}$. The area of the region outside $X_{20}$ but inside $X_{21}$ can be written as $a\pi + b$, for integers $a$ and $b$. Compute $100a + b$.

Answer: 4112
user: Let $N$ be the number of ways of distributing $52$ identical balls into $4$ distinguishable boxes such that no box is empty and the difference between the number of balls in any two of the boxes is not a multiple of $6$. If $N = 100a + b$, where $a, b$ are positive integers less than $100$, find $a + b$.

Answer: 81
user: Let $ABCD$ be a cyclic quadrilateral, with $AB = 7$, $BC = 11$, $CD = 13$, and $DA = 17$. Let the incircle of $\triangle ABD$ touch $BD$ at $R$ and the incircle of $\triangle CBD$ touch $BD$ at $S$. What is $RS$?

Answer: 4
user: Let $a_1, a_2, \ldots$ be an infinite sequence of integers such that $a_i$ divides $a_{i+1}$ for all $i \geq 1$. Let $b_i$ be the remainder when $a_i$ is divided by $210$. What is the maximal number of distinct terms in the sequence $b_1, b_2, \ldots$?

Answer: 16
user: Let $a$, $b$, $c$, and $d$ be real numbers such that \[ a^2 + 3b^2 + \frac{c^2 + 3d^2}{2} = a + b + c + d - 1. \] Find $1000a + 100b + 10c + d$.

Answer: 527