meetforever/GRPO_CL huggingface dataset:
user: Let $\triangle ABC$ be a triangle with $BC = 4$, $CA= 5$, $AB= 6$, and let $O$ be the circumcenter of $\triangle ABC$. Let $O_b$ and $O_c$ be the reflections of $O$ about lines $CA$ and $AB$ respectively. Suppose $BO_b$ and $CO_c$ intersect at $T$, and let $M$ be the midpoint of $BC$. Given that $MT^2 = \frac{p}{q}$ for some coprime positive integers $p$ and $q$, find $p+q$.
Answer: 23
user: Jerry's favorite number is $97$. He knows all kinds of interesting facts about $97$:

- $97$ is the largest two-digit prime.
- Reversing the order of its digits results in another prime.
- There is only one way in which $97$ can be written as a difference of two perfect squares.
- There is only one way in which $97$ can be written as a sum of two perfect squares.
- $\frac{1}{97}$ has exactly $96$ digits in the smallest repeating block of its decimal expansion.
- Jerry blames the sock gnomes for the theft of exactly $97$ of his socks.

A repunit is a natural number whose digits are all $1$. For instance, $1$, $11$, $111$, $1111$, $\ldots$ are the four smallest repunits. How many digits are there in the smallest repunit that is divisible by $97$?
Answer: 96
user: Determine the number of ordered quadruples $(a,b,c,d)$ of integers, where $a,b,c,d$ are integers between $1$ and $3$ inclusive, such that the pairs $(a,b)$ and $(c,d)$ form a basis of the coordinate plane. A pair $(a,b)$ and $(c,d)$ form a basis if $ad \neq bc$.
Answer: 66
user: The summation \( \sum_{k=1}^{360} \frac{1}{k \sqrt{k+1} + (k+1)\sqrt{k}} \) is the ratio of two relatively prime positive integers \( m \) and \( n \). Find \( m + n \).
Answer: 37
user: Let $A$ be the set of positive integers that have no prime factors other than $2$, $3$, or $5$. The infinite sum \[ \frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \frac{1}{6} + \frac{1}{8} + \frac{1}{9} + \frac{1}{10} + \frac{1}{12} + \frac{1}{15} + \frac{1}{16} + \frac{1}{18} + \frac{1}{20} + \cdots \] of the reciprocals of the elements of $A$ can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$?

**(A) 16**  **(B) 17**  **(C) 19**  **(D) 23**  **(E) 36**
Answer: 19
user: The trapezoid below has bases with lengths 7 and 17, and an area of 120. Find the difference in the areas of the two triangles formed by the diagonals of the trapezoid.
Answer: 50
user: We say that an ordered pair $(a,b)$ of positive integers with $a > b$ is square-ish if both $a + b$ and $a - b$ are perfect squares. For example, $(17,8)$ is square-ish because $17 + 8 = 25$ and $17 - 8 = 9$ are both perfect squares. How many square-ish pairs $(a,b)$ with $a + b < 100$ are there?
Answer: 16
user: Suppose $a$ is a real number such that $\sin(\pi \cdot \cos a) = \cos(\pi \cdot \sin a)$. Evaluate $35 \sin^2(2a) + 84 \cos^2(4a)$.
Answer: 21
user: Let $\underline{xyz}$ represent the three-digit number with hundreds digit $x$, tens digit $y$, and units digit $z$. Similarly, let $\underline{yz}$ represent the two-digit number with tens digit $y$ and units digit $z$. How many three-digit numbers $\underline{abc}$, none of whose digits are 0, are there such that $\underline{ab} > \underline{bc} > \underline{ca}$?
Answer: 120
user: Find the largest positive integer $N$ such that the number of integers in the set $\{1, 2, \dots, N\}$ divisible by 3 is equal to the number of integers divisible by 5 or 7 (or both).
Answer: 65
user: Find all positive integers $n$ such that:
$$n = a^2 + b^2 + c^2 + d^2,$$
where $a < b < c < d$ are the smallest divisors of $n$.
Answer: 130
user: A spider is walking on the boundary of an equilateral triangle $\triangle{ABC}$, with vertices labeled in counterclockwise order, starting at vertex $A$. Each second, she moves to one of her two adjacent vertices with equal probability. The windiness of a path that starts and ends at $A$ is the net number of counterclockwise revolutions made. For example, the windiness of the path $ABCA$ is $1$, and the windiness of the path $ABCACBACBA$ is $-1$. What is the remainder modulo $1000$ of the sum of the squares of the windiness values taken over all possible paths that end back at vertex $A$ after $2025$ seconds?
Answer: 625
user: Compute the sum of the digits of $101^6$.
Answer: 28
user: Find the sum of all values of $a$ such that there are positive integers $a$ and $b$ satisfying $(a - b)\sqrt{ab} = 2016$.
Answer: 209
user: Circle $\Gamma$ is centered at $(0, 0)$ in the plane with radius $2022\sqrt{3}$. Circle $\Omega$ is centered on the $x$-axis, passes through the point $A = (6066, 0)$, and intersects $\Gamma$ orthogonally at the point $P = (x, y)$ with $y > 0$. If the length of the minor arc $AP$ on $\Omega$ can be expressed as $\frac{m\pi}{n}$ for relatively prime positive integers $m, n$, find $m + n$.
Answer: 1349
user: We take $100$ consecutive natural numbers $a_1, a_2, \ldots, a_{100}$. Determine the last two digits of the number $a_1^8 + a_2^8 + \ldots + a_{100}^8$.
Answer: 30
user: Consider the polynomial $f(x) = ax^2 + bx + c$, with degree less than or equal to 2.

When $f$ varies subject to the constraints $f(0) = 0$ and $f(2) = 2$, find the minimum value of $S = \int_0^2 |f'(x)|\, dx$.
Answer: 2
user: Find the sum of all the real values of $x$ satisfying the equation:
\[(x+\frac{1}{x}-17)^2 = x + \frac{1}{x} + 17.\]
Answer: 35
user: Five girls and five boys randomly sit in ten seats that are equally spaced around a circle. The probability that there is at least one diameter of the circle with two girls sitting on opposite ends of the diameter is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
Answer: 118
user: How many positive integers $N$ in the segment $\left[10, 10^{20} \right]$ are such that if all their digits are increased by 1 and then multiplied, the result is $N+1$?
Answer: 171
user: The product of a million whole numbers is equal to one million. What can be the greatest possible value of the sum of these numbers?
Answer: 1999999
user: The leading coefficient $a$ in the quadratic polynomial $P(x) = ax^2 + bx + c$ is greater than $100$. Determine the maximum number of integer values for $x$ such that $|P(x)| < 50$.
Answer: 2
user: A grasshopper starts at the origin in the coordinate plane and makes a sequence of hops. Each hop has a length of $5$, and after each hop, the grasshopper is at a point whose coordinates are both integers. Thus, there are $12$ possible locations for the grasshopper after the first hop. What is the smallest number of hops needed for the grasshopper to reach the point $(2021,2021)$?
Answer: 578
user: Let $\ell$ be a line and let points $A$, $B$, $C$ lie on $\ell$ so that $AB = 7$ and $BC = 5$. Let $m$ be the line through $A$ perpendicular to $\ell$. Let $P$ lie on $m$. Compute the smallest possible value of $PB + PC$.
Answer: 9
user: A clueless ant makes the following route: starting at point $A$, it goes $1$ cm north, then $2$ cm east, then $3$ cm south, then $4$ cm west, immediately $5$ cm north, continues $6$ cm east, and so on, finally $41$ cm north and ends at point $B$. Calculate the distance between $A$ and $B$ in a straight line.
Answer: 29
user: The trapezoid $ABCD$ has bases with lengths $7$ and $17$ and an area of $120$. Find the difference of the areas of $\triangle ACD$ and $\triangle CDB$.
Answer: 0
user: Alice and Bob are independently trying to figure out a secret password to Cathy’s bitcoin wallet. Both of them have already figured out that:
- It is a $4$-digit number whose first digit is $5$.
- It is a multiple of $9$.
- The larger number is more likely to be a password than a smaller number.

Moreover, Alice figured out the second and the third digits of the password, and Bob figured out the third and the fourth digits. They told this information to each other but not the actual digits. After that, the conversation followed:

Alice: "I have no idea what the number is."
Bob: "I have no idea too."

After that, both of them knew which number they should try first. Identify this number.
Answer: 5940
user: Squares $ABCD$ and $AEFG$, each with side length $12$, overlap such that $\triangle AED$ is an equilateral triangle, as shown in the diagram. The area of the region that lies within both squares, which is shaded in the diagram, is given by $m\sqrt{n}$, where $m$ and $n$ are positive integers, and $n$ is not divisible by the square of any prime. Determine the value of $m + n$.
Answer: 51
user: The sum \(\sum_{k=3}^{\infty} \frac{1}{k(k^4-5k^2+4)^2}\) is equal to \(\frac{m^2}{2n^2}\), where \(m\) and \(n\) are relatively prime positive integers. Find \(m+n\).
Answer: 49
user: Carl, James, Saif, and Ted play several games of two-player For The Win on the Art of Problem Solving website. If, among these games, Carl wins $5$ and loses $0$, James wins $4$ and loses $2$, Saif wins $1$ and loses $6$, and Ted wins $4$, how many games does Ted lose?
Answer: 6
user: Let $a_1$, $a_2$, $\cdots$ be a sequence such that $a_1 = a_2 = \frac{1}{5}$, and for $n \ge 3$,  
\[a_n = \frac{a_{n-1} + a_{n-2}}{1 + a_{n-1}a_{n-2}}.\]  
Find the smallest integer $n$ such that $a_n > 1 - 5^{-2022}$.
Answer: 21
user: Two people $A$ and $B$ start from the same place at the same time to travel around a circular track of length $100$ m in opposite directions. Initially, $B$ moves more slowly than $A$ until they meet. After meeting, $B$ doubles his speed and next meets $A$ at the starting point. Let $d$ m be the distance traveled by $B$ before he met $A$ for the first time after leaving the starting point. Find the integer closest to $d$.
Answer: 41
user: In a rectangular $57 \times 57$ grid of cells, $k$ of the cells are colored black. What is the smallest positive integer $k$ such that there must exist a rectangle, with sides parallel to the edges of the grid, that has its four vertices at the center of distinct black cells?
Answer: 457
user: Jonah recently harvested a large number of lychees and wants to split them into groups. Unfortunately, for all $n$ where $3 \leq n \leq 8$, when the lychees are distributed evenly into $n$ groups, $n-1$ lychees remain. What is the smallest possible number of lychees that Jonah could have?
Answer: 839
user: Daeun draws a unit circle centered at the origin and inscribes within it a regular hexagon $ABCDEF$. Then Dylan chooses a point $P$ within the circle of radius $2$ centered at the origin. Let $M$ be the maximum possible value of $|PA| \cdot |PB| \cdot |PC| \cdot |PD| \cdot |PE| \cdot |PF|$, and let $N$ be the number of possible points $P$ for which this maximal value is obtained. Find $M + N^2$.
Answer: 101
user: A right rectangular prism has integer side lengths $a$, $b$, and $c$. If $\text{lcm}(a,b)=72$, $\text{lcm}(a,c)=24$, and $\text{lcm}(b,c)=18$, what is the sum of the minimum and maximum possible volumes of the prism?
Answer: 3024
user: For a positive integer $n$, let $f(n)$ be the integer formed by reversing the digits of $n$ (and removing any leading zeroes). For example, $f(14172) = 27141$. Define a sequence of numbers $\{a_n\}_{n \ge 0}$ by $a_0 = 1$ and for all $i \ge 0$, $a_{i+1} = 11a_i$ or $a_{i+1} = f(a_i)$. How many possible values are there for $a_8$?
Answer: 13
user: Regular hexagon $ABCDEF$ has side length $2$. Points $M$ and $N$ lie on $BC$ and $DE$, respectively. Find the minimum possible value of $(AM + MN + NA)^2$.
Answer: 84
user: Let $n$ be a fixed positive integer. For integer $m$ satisfying $|m| \le n$, define 
\[ S_m = \sum_{\substack{i-j=m \\ 0 \le i,j \le n}} \frac{1}{2^{i+j}}. \]
Then 
\[ \lim_{n\to \infty} \left( S_{-n}^2 + S_{-n+1}^2 + \ldots + S_n^2 \right) \]
can be expressed in the form $\frac{p}{q}$ for relatively prime positive integers $p, q$. Compute $100p + q$.
Answer: 8027
user: Let $S$ be the smallest subset of the integers with the property that $0 \in S$ and for any $x \in S$, we have $3x \in S$ and $3x + 1 \in S$. Determine the number of non-negative integers in $S$ less than $2008$.
Answer: 128
user: Determine the smallest integer $A$ such that there exist consecutive integers, including $A$, which sum to 2014. An integer $A$ with this property is called "yummy."
Answer: -2013
user: Let $\overline{ABCD}$ be a 4-digit number. What is the smallest possible positive value of $\overline{ABCD} - \overline{DCBA}$?
Answer: 90
user: Let $x$ be the first term in the sequence $31, 331, 3331, \ldots$ which is divisible by $17$. How many digits long is $x$?
Answer: 9
user: Suppose $w, x, y, z$ satisfy the following equations:
\[
\begin{align*}
w + x + y + z &= 25, \\
wx + wy + wz + xy + xz + yz &= 2y + 2z + 193
\end{align*}
\]
The largest possible value of $w$ can be expressed in lowest terms as $\frac{w_1}{w_2}$ for some integers $w_1, w_2 > 0$. Find $w_1 + w_2$.
Answer: 27
user: Let $r_k$ denote the remainder when $\binom{127}{k}$ is divided by $8$. Compute $r_1 + 2r_2 + 3r_3 + \cdots + 63r_{63}$.
Answer: 2016
user: Given a positive integer $x > 1$ with $n$ divisors, define $f(x)$ to be the product of the smallest $\lceil \frac{n}{2} \rceil$ divisors of $x$. Let $a$ be the least value of $x$ such that $f(x)$ is a multiple of $x$, and $b$ be the least value of $n$ such that $f(y)$ is a multiple of $y$ for some $y$ that has exactly $n$ factors. Compute $a + b$.
Answer: 31
user: How many integers of the form $n^{2023-n}$ are perfect squares, where $n$ is a positive integer between $1$ and $2023$ inclusive?
Answer: 1033
user: Let $P$ be a point on the graph of the function $y = x + \frac{2}{x}$ for $x > 0$. The lines $PA$ and $PB$ are perpendicular to the lines $y = x$ and $x = 0$, respectively, with $A$ and $B$ being the feet of these perpendiculars. Find the value of $\overrightarrow{PA} \cdot \overrightarrow{PB}$.
Answer: -1
user: Let $f(x) = x^3 - 7x^2 + 16x - 10$. As $x$ ranges over all integers, find the sum of distinct prime values taken on by $f(x)$.
Answer: 2
user: There are $522$ people at a beach, each of whom owns a cat, a dog, both, or neither. If $20\%$ of cat-owners also own a dog, $70\%$ of dog-owners do not own a cat, and $50\%$ of people who don’t own a cat also don’t own a dog, how many people own neither type of pet?
Answer: 126
user: Given a cube, on each edge, write a number, either $1$ or $-1$. For each face of the cube, multiply the four numbers on the edges of this face and write the product on this face. Finally, add all the eighteen numbers that were written on the edges and faces of the cube.

What is the smallest possible sum that can be obtained?
Answer: -12
user: The following sequence lists all the positive rational numbers that do not exceed $\frac{1}{2}$ by first listing the fraction with denominator 2, followed by the one with denominator 3, followed by the two fractions with denominator 4 in increasing order, and so forth. The sequence is as follows:
\[
\frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{2}{4}, \frac{1}{5}, \frac{2}{5}, \frac{1}{6}, \frac{2}{6}, \frac{3}{6}, \frac{1}{7}, \frac{2}{7}, \frac{3}{7}, \cdots.
\]
Let $m$ and $n$ be relatively prime positive integers such that the $2012^{\text{th}}$ fraction in the list is equal to $\frac{m}{n}$. Find $m+n$.
Answer: 61
user: The function $f(n)$ is defined on the positive integers and takes non-negative integer values. It is given that $f(2)=0$, $f(3)>0$, and $f(9999)=3333$. For all integers $m$ and $n$, the following condition holds:
\[
f(m+n)-f(m)-f(n)=0 \text{ or } 1.
\]
Determine the value of $f(1982)$.
Answer: 660
user: Let $M = \{1,2,\cdots , 10\}$, and let $T$ be a set of 2-element subsets of $M$. For any two different elements $\{a,b\}, \{x,y\}$ in $T$, the integer $(ax+by)(ay+bx)$ is not divisible by 11. Find the maximum size of $T$.
Answer: 25
user: The sequence $(a_n)$ is defined by the following initial conditions and recurrence relations:

- Initial conditions: 
  \[ a_1 = 1, \quad a_2 = 0 \]

- Recurrence relations for $k \in \mathbb{N}$:
  \[ a_{2k+1} = a_k + a_{k+1}, \quad a_{2k+2} = 2a_{k+1} \]

Find the value of $a_m$ for $m = 2^{19} + 91$.
Answer: 91
user: Find the value of $x$ where the graph of \( y = \log_3(\sqrt{x^2+729}+x) - 2\log_3(\sqrt{x^2+729}-x) \) crosses the $x$-axis.
Answer: 36
user: Let $T$ be the set of all positive integer divisors of $2004^{100}$. What is the largest possible number of elements of a subset $S$ of $T$ such that no element in $S$ divides any other element in $S$?
Answer: 10201
user: Sets $A$, $B$, and $C$ satisfy the following conditions:  
- $|A| = 92$  
- $|B| = 35$  
- $|C| = 63$  
- $|A \cap B| = 16$  
- $|A \cap C| = 51$  
- $|B \cap C| = 19$  

Compute the number of possible values of $|A \cap B \cap C|$.
Answer: 10
user: Let $a, b, c$ be positive integers such that $a + 2b + 3c = 100$. Find the greatest value of $M = abc$.
Answer: 6171
user: Olya wrote fractions of the form $\frac{1}{n}$ on cards, where $n$ represents all possible divisors of the number $6^{100}$ (including the unit and the number itself). She arranged these cards in some order. 

After arranging, she noted the number on the first card, then the sum of the numbers on the first and second cards, then the sum of the numbers on the first three cards, and so on, until finally, she recorded the sum of the numbers on all the cards. Each sum was recorded on the board as an irreducible fraction. 

What is the least number of different denominators that could appear on the numbers on the board?
Answer: 2
user: Let's consider all possible natural seven-digit numbers, in the decimal notation of which the digits $1$, $2$, $3$, $4$, $5$, $6$, $7$ are used once each. Let's number these numbers in ascending order. What number will be the $1995^{th}$?
Answer: 3651427
user: Find the positive integer $n$ such that:
\[
\underbrace{f(f(\cdots f}_{2013 \ f\text{'s}}(n)\cdots ))=2014^2+1
\]
where $f(n)$ denotes the $n$th positive integer which is not a perfect square.
Answer: 1015057
user: A triangulation of a polygon is a subdivision of the polygon into triangles meeting edge to edge, with the property that the set of triangle vertices coincides with the set of vertices of the polygon. Adam randomly selects a triangulation of a regular $180$-gon. Then, Bob selects one of the $178$ triangles in this triangulation. The expected number of $1^\circ$ angles in this triangle can be expressed as $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers. Compute $100a + b$.
Answer: 9089
user: For a positive integer $n > 1$, let $g(n)$ denote the largest positive proper divisor of $n$ and $f(n) = n - g(n)$. For example, $g(10) = 5$, $f(10) = 5$ and $g(13) = 1$, $f(13) = 12$. Let $N$ be the smallest positive integer such that $f(f(f(N))) = 97$. Find the largest integer not exceeding $\sqrt{N}$.
Answer: 19
user: For positive integers $i$ and $j$, define $d(i,j)$ as follows:  
- $d(1,j) = 1$ for all $j$  
- $d(i,1) = 1$ for all $i$  
- For $i, j > 1$, $d(i,j) = d(i-1,j) + d(i,j-1) + d(i-1,j-1)$

Compute the remainder when $d(3,2016)$ is divided by $1000$.
Answer: 481
user: Altitudes $BE$ and $CF$ of acute triangle $ABC$ intersect at $H$. Suppose that the altitudes of triangle $EHF$ concur on line $BC$. If $AB=3$ and $AC=4$, then $BC^2=\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers. Compute $100a+b$.
Answer: 33725
user: Let $p$ be a polynomial with degree less than $4$ such that $p(x)$ attains a maximum at $x = 1$. If $p(1) = p(2) = 5$, find $p(10)$.
Answer: 5
user: Consider the sequences of six positive integers $a_1,a_2,a_3,a_4,a_5,a_6$ with the properties that $a_1=1$, and if for some $j > 1$, $a_j = m > 1$, then $m-1$ appears in the sequence $a_1,a_2,\ldots,a_{j-1}$. Such sequences include $1,1,2,1,3,2$ and $1,2,3,1,4,1$ but not $1,2,2,4,3,2$. How many such sequences of six positive integers are there?
Answer: 203
user: Three spheres are all externally tangent to a plane and to each other. Suppose that the radii of these spheres are $6$, $8$, and $10$. The tangency points of these spheres with the plane form the vertices of a triangle. Determine the largest integer that is smaller than the perimeter of this triangle.
Answer: 47
user: The sides $x$ and $y$ of a scalene triangle satisfy the equation $x + \frac{2\Delta}{x} = y + \frac{2\Delta}{y}$, where $\Delta$ is the area of the triangle. Given that $x = 60$ and $y = 63$, determine the length of the largest side of the triangle.
Answer: 87
user: Define the function $f: \mathbb{R} \setminus \{-1,1\} \to \mathbb{R}$ as follows:
\[ f(x) = \sum_{a,b=0}^{\infty} \frac{x^{2^a3^b}}{1-x^{2^{a+1}3^{b+1}}} . \]
Suppose that $f(y) - f\left(\frac{1}{y}\right) = 2016$. Then, express $y$ in its simplest form as $\frac{p}{q}$. Find $p+q$.
Answer: 4033
user: If $a$ and $b$ are positive integers such that $3\sqrt{2+\sqrt{2+\sqrt{3}}}=a\cos\frac{\pi}{b}$, find $a+b$.
Answer: 30
user: 21 Savage has a 12-car garage, with a row of spaces numbered 1, 2, 3, \ldots, 12. How many ways can he choose 6 of them to park his 6 identical cars in, if no 3 spaces with consecutive numbers may be all occupied?
Answer: 357
user: Let $S = \{1, 2, \ldots, 8\}$. How many ways are there to select two disjoint subsets of $S$?
Answer: 6561
user: Given a function $p(x) = ax^5 + bx^4 + cx^3 + dx^2 + ex + f$, where each coefficient $a, b, c, d, e,$ and $f$ is either $1$ or $-1$. If $p(2) = 11$, what is the value of $p(3)$?
Answer: 142
user: In triangle $ABC$, given $AB = 2$, $AC = 1 + \sqrt{5}$, and $\angle CAB = 54^{\circ}$. Suppose $D$ lies on the extension of $AC$ through $C$ such that $CD = \sqrt{5} - 1$. If $M$ is the midpoint of $BD$, determine the measure of $\angle ACM$ in degrees.
Answer: 63
user: In the diagram below, \( \overline{AB} \) and \( \overline{CD} \) are parallel, \( \angle BXY = 45^\circ \), \( \angle DZY = 25^\circ \), and \( XY = YZ \). What is the degree measure of \( \angle YXZ \)?
Answer: 55
user: Let $f(x) = x^{2} + ax + b$. If for all nonzero real $x$:
\[ f\left(x + \frac{1}{x}\right) = f\left(x\right) + f\left(\frac{1}{x}\right) \]
and the roots of $f(x) = 0$ are integers, what is the value of $a^{2} + b^{2}$?
Answer: 13
user: Given a triangle $ABC$ with angles $\angle A = 60^{\circ}$, $\angle B = 75^{\circ}$, $\angle C = 45^{\circ}$, let $H$ be its orthocenter, and $O$ be its circumcenter. Let $F$ be the midpoint of side $AB$, and $Q$ be the foot of the perpendicular from $B$ onto $AC$. Denote by $X$ the intersection point of the lines $FH$ and $QO$. Suppose the ratio of the length of $FX$ and the circumradius of the triangle is given by $\frac{a + b \sqrt{c}}{d}$, then find the value of $1000a + 100b + 10c + d$.
Answer: 1132
user: Let $\triangle ABC$ be a triangle with side lengths $AB = 9$, $BC = 10$, $CA = 11$. Let $O$ be the circumcenter of $\triangle ABC$. Denote $D = AO \cap BC$, $E = BO \cap CA$, $F = CO \cap AB$. If $\frac{1}{AD} + \frac{1}{BE} + \frac{1}{FC}$ can be written in simplest form as $\frac{a \sqrt{b}}{c}$, find $a + b + c$.
Answer: 43
user: You have two blackboards, $A$ and $B$. You need to write some integers on them, where each integer is greater than or equal to $2$ and less than or equal to $20$. The condition is that each number on blackboard $A$ must be co-prime with each number on blackboard $B$. Determine the maximum possible value of the product of the number of integers written on $A$ and the number of integers written on $B$.
Answer: 65
user: We know that $201$ and $9$ give the same remainder when divided by $24$. What is the smallest positive integer $k$ such that $201+k$ and $9+k$ give the same remainder when divided by $24+k$?
Answer: 8
user: The diagram shows two equilateral triangles with side length $4$ mounted on two adjacent sides of a square, also with side length $4$. The distance between the two vertices marked $A$ and $B$ can be written as $\sqrt{m}+\sqrt{n}$ for two positive integers $m$ and $n$. Find $m + n$.
Answer: 32
user: Calculate \(1^2 - 2^2 + 3^2 - 4^2 + \ldots - 2018^2 + 2019^2\).
Answer: 2039190
user: Let $\triangle ABC$ be a triangle with $AB = 10$ and $AC = 11$. Let $I$ be the center of the inscribed circle of $\triangle ABC$. If $M$ is the midpoint of $AI$ such that $BM = BC$ and $CM = 7$, then $BC$ can be expressed in the form $\frac{\sqrt{a}-b}{c}$ where $a$, $b$, and $c$ are positive integers. Find $a+b+c$.
Answer: 622
user: Determine the maximum integer $n$ such that for each positive integer $k \le \frac{n}{2}$, there exist two positive divisors of $n$ with a difference of $k$.
Answer: 24