meetforever/GRPO_CL huggingface dataset:
user: One integer was removed from the set $S=\{1, 2, 3, \ldots, n\}$ of the integers from $1$ to $n$. The arithmetic mean of the other integers of $S$ is equal to $\frac{163}{4}$. What integer was removed?
Answer: 61
user: Find the number of positive integer divisors of $12!$ that leave a remainder of $1$ when divided by $3$.
Answer: 66
user: Find the smallest positive integer solution to the equation $2^{2^k} \equiv k \pmod{29}$.
Answer: 24
user: Given three equilateral triangles $\triangle HOW$, $\triangle BOW$, and $\triangle DAH$ in a plane:
- $WO = 7$
- $AH = 2$

It is also given that the points $D$, $A$, and $B$ are collinear in that order. 

Find the length of $BA$.
Answer: 11
user: Let $r$ be a real number such that $\sqrt[3]{r} - \frac{1}{\sqrt[3]{r}} = 2$. Find $r^3 - \frac{1}{r^3}$.
Answer: 2786
user: A cyclic quadrilateral $ABCD$ has side lengths $AB = 3$, $BC = AD = 5$, and $CD = 8$. The radius of its circumcircle can be expressed in the form $\frac{a\sqrt{b}}{c}$, where $a$, $b$, $c$ are positive integers, $a$ and $c$ are relatively prime, and $b$ is not divisible by the square of any prime. Find $a + b + c$.
Answer: 13
user: Let $ABC$ be a triangle with vertices located at the centers of mass of Vincent Huang's house, Tristan Shin's house, and Edward Wan's house; assume the three are not collinear. Let $N = 2017$, and define the $A$-antipodes to be the points $A_1, \dots, A_N$ on segment $BC$ such that $BA_1 = A_1A_2 = \cdots = A_{N-1}A_N = A_NC$. Similarly, define the $B$-antipodes and $C$-antipodes. A line $\ell_A$ through $A$ is called a qevian if it passes through an $A$-antipode. Similarly, define qevians through $B$ and $C$. Compute the number of ordered triples $(\ell_A, \ell_B, \ell_C)$ of concurrent qevians through $A$, $B$, and $C$, respectively.
Answer: 6049
user: Among the four-digit numbers greater than $2022$ that are divisible by $3$, find the smallest integer whose four digits consist of only two different kinds of numbers.
Answer: 2112
user: Let $n$ be a positive integer. What is the smallest sum of digits that $5^n + 6^n + 2022^n$ can take?
Answer: 8
user: What is the greatest number of balls with a radius of $\frac{1}{2}$ that can be placed within a rectangular box of size $10 \times 10 \times 1$?
Answer: 106
user: For a positive integer $n$, let $d(n)$ be the number of positive divisors of $n$. What is the smallest positive integer $n$ such that \[ \sum_{t \mid n} d(t)^3 \] is divisible by $35$?
Answer: 864
user: For polynomials $P(x) = a_nx^n + \cdots + a_0$, let $f(P) = a_n \cdots a_0$ be the product of the coefficients of $P$. The polynomials $P_1, P_2, P_3, Q$ satisfy $P_1(x) = (x-a)(x-b)$, $P_2(x) = (x-a)(x-c)$, $P_3(x) = (x-b)(x-c)$, $Q(x) = (x-a)(x-b)(x-c)$ for some complex numbers $a, b, c$. Given $f(Q) = 8$, $f(P_1) + f(P_2) + f(P_3) = 10$, and $abc > 0$, find the value of $f(P_1)f(P_2)f(P_3)$.
Answer: 32
user: Compute the sum of the distinct prime factors of $10101$.
Answer: 60
user: Sequences $a_n$ and $b_n$ are defined for all positive integers $n$ such that $a_1 = 5$, $b_1 = 7$,  

$$a_{n+1} = \frac{\sqrt{(a_n+b_n-1)^2+(a_n-b_n+1)^2}}{2},$$

and 

$$b_{n+1} = \frac{\sqrt{(a_n+b_n+1)^2+(a_n-b_n-1)^2}}{2}.$$  

How many integers $n$ from 1 to 1000 satisfy the property that $a_n, b_n$ form the legs of a right triangle with a hypotenuse that has integer length?
Answer: 24
user: Let $\mathcal{S} = \sum_{i=1}^{\infty}\left(\prod_{j=1}^i \dfrac{3j - 2}{12j}\right).$ Then $(\mathcal{S} + 1)^3 = \frac{m}{n}$ with $m$ and $n$ coprime positive integers. Find $10m + n$.
Answer: 43
user: Let $\alpha = \cos^{-1} \left( \frac{3}{5} \right)$ and $\beta = \sin^{-1} \left( \frac{3}{5} \right)$.  
\[ \sum_{n=0}^{\infty}\sum_{m=0}^{\infty} \frac{\cos(\alpha n +\beta m)}{2^n3^m} \]
can be written as $\frac{A}{B}$ for relatively prime positive integers $A$ and $B$. Find $1000A + B$.
Answer: 15013
user: What is the smallest positive integer $n$ such that $20 \equiv n^{15} \pmod{29}$?
Answer: 20
user: Given two natural numbers $w$ and $n$, the tower of $n$ $w$'s is the natural number $T_n(w)$ defined by

\[ T_n(w) = w^{w^{\cdots^{w}}},\]

with $n$ $w$'s on the right side. More precisely, $T_1(w) = w$ and $T_{n+1}(w) = w^{T_n(w)}$. For example, $T_3(2) = 2^{2^2} = 16$, $T_4(2) = 2^{16} = 65536$, and $T_2(3) = 3^3 = 27$. Find the smallest tower of $3$'s that exceeds the tower of $1989$ $2$'s. In other words, find the smallest value of $n$ such that $T_n(3) > T_{1989}(2)$. Justify your answer.
Answer: 1988
user: How many ways are there to arrange the array of positive integers such that:
- The number $2012$ is in the first position.
- The number $1$ is in the last position.

Note: Each number following $n$ must be a positive integer less than $\sqrt{n}$.
Answer: 201
user: For her daughter's $12^{\text{th}}$ birthday, Ingrid decides to bake a dodecagon pie in celebration. Unfortunately, the store does not sell dodecagon-shaped pie pans, so Ingrid bakes a circular pie first and then trims off the sides in a way such that she gets the largest regular dodecagon possible. If the original pie was $8$ inches in diameter, the area of pie that she has to trim off can be represented in square inches as $a\pi - b$ where $a, b$ are integers. What is $a + b$?
Answer: 64
user: How many of the numbers $1 \cdot 2 \cdot 3$, $2 \cdot 3 \cdot 4$, ..., $2020 \cdot 2021 \cdot 2022$ are divisible by $2020$?
Answer: 27
user: Find the circumradius of the triangle with side lengths $104$, $112$, and $120$.
Answer: 65
user: Positive integers $a < b < c$ are such that $b + a$ is a multiple of $b - a$ and $c + b$ is a multiple of $c - b$. If $a$ is a $2011$-digit number and $b$ is a $2012$-digit number, exactly how many digits does $c$ have?
Answer: 2013
user: A coloring of all plane points with coordinates belonging to the set $S=\{0,1,\ldots,99\}$ into red and white colors is said to be critical if for each $i,j \in S$ at least one of the four points $(i,j), (i + 1,j), (i,j + 1)$, and $(i + 1, j + 1)$ $(99 + 1 \equiv 0)$ is colored red. Find the maximal possible number of red points in a critical coloring which loses its property after recoloring of any red point into white.
Answer: 5000
user: There are $1000$ students in a school. Every student has exactly $4$ friends. A group of three students $\{A,B,C\}$ is said to be a friendly triplet if any two students in the group are friends. Determine the maximal possible number of friendly triplets.
Answer: 2000
user: How many positive integers between 1 and 400 (inclusive) have exactly 15 positive integer factors?
Answer: 3
user: Let $f(x)$ be a function such that $f(1) = 1234$, $f(2) = 1800$, and $f(x) = f(x-1) + 2f(x-2) - 1$ for all integers $x$. Evaluate the number of divisors of 
\[ \sum_{i=1}^{2022} f(i) \]
Answer: 8092
user: Calculate the value of $\sqrt{\frac{11^4+100^4+111^4}{2}}$ and provide your answer as an integer.
Answer: 11221
user: Find the number of ordered pairs of integers \((a, b)\) that satisfy the inequality

\[
1 < a < b+2 < 10.
\]
Answer: 28
user: Mr. DoBa likes to listen to music occasionally while he does his math homework. When he listens to classical music, he solves one problem every $3$ minutes. When he listens to rap music, however, he only solves one problem every $5$ minutes. Mr. DoBa listens to a playlist comprised of $60\%$ classical music and $40\%$ rap music. Each song is exactly $4$ minutes long. Suppose that the expected number of problems he solves in an hour does not depend on whether or not Mr. DoBa is listening to music at any given moment, and let $m$ be the average number of problems Mr. DoBa solves per minute when he is not listening to music. Determine the value of $1000m$.
Answer: 280
user: $q$ is the smallest rational number having the following properties:

1. $q > \frac{31}{17}$

2. When $q$ is written in its reduced form $\frac{a}{b}$, then $b < 17$.

Find $a + b$.
Answer: 17
user: Consider the $4 \times 4$ array of $16$ dots, shown below.

Counting the number of squares whose vertices are among the $16$ dots and whose sides are parallel to the sides of the grid, we find that there are nine $1 \times 1$ squares, four $2 \times 2$ squares, and one $3 \times 3$ square, for a total of $14$ squares. We delete a number of these dots. What is the minimum number of dots that must be deleted so that each of the $14$ squares is missing at least one vertex?
Answer: 4
user: Jeffrey rolls three fair six-sided dice and records their results. The probability that the mean of these three numbers is greater than the median of these three numbers can be expressed as \( \frac{m}{n} \) for relatively prime positive integers \( m \) and \( n \). Compute \( m+n \).
Answer: 101
user: Let \(x\) and \(y\) be positive real numbers satisfying the following system of equations:
\[
\begin{cases}
\sqrt{x}\left(2 + \dfrac{5}{x+y}\right) = 3 \\\\
\sqrt{y}\left(2 - \dfrac{5}{x+y}\right) = 2
\end{cases}
\]
Find the maximum value of \(x + y\).
Answer: 5
user: A $3 \times 3$ grid is to be painted with three colors: red, green, and blue, such that:
1. No two squares that share an edge are the same color.
2. No two corner squares on the same edge of the grid have the same color.

For example, the upper-left and bottom-left squares cannot both be red, as that would violate condition (ii). In how many ways can this be done? Note that rotations and reflections are considered distinct colorings.
Answer: 24
user: Portraits of famous scientists hang on a wall. The scientists lived between 1600 and 2008, and none of them lived longer than 80 years. Vasya multiplied the years of birth of these scientists, and Petya multiplied the years of their death. Petya's result is exactly $\frac{5}{4}$ times greater than Vasya's result. What is the minimum number of portraits that can be on the wall?
Answer: 5
user: \( \triangle ABC \) has \( AB = 4 \) and \( AC = 6 \). Let point \( D \) be on line \( AB \) so that \( A \) is between \( B \) and \( D \). Let the angle bisector of \( \angle BAC \) intersect line \( BC \) at \( E \), and let the angle bisector of \( \angle DAC \) intersect line \( BC \) at \( F \). Given that \( AE = AF \), find the square of the circumcircle's radius' length.
Answer: 13
user: Let $ABC$ be a triangle with $AB = 4024$, $AC = 4024$, and $BC = 2012$. The reflection of line $AC$ over line $AB$ meets the circumcircle of $\triangle ABC$ at a point $D \neq A$. Find the length of segment $CD$.
Answer: 3521
user: Suppose 32 objects are placed along a circle at equal distances. In how many ways can 3 objects be chosen from among them so that no two of the three chosen objects are adjacent nor diametrically opposite?
Answer: 3616
user: You have 2020 piles of coins in front of you. The first pile contains 1 coin, the second pile contains 2 coins, the third pile contains 3 coins, and so on. Therefore, the 2020th pile contains 2020 coins. Guess a positive integer $k$, such that the piles contain at least $k$ coins. Take away exactly $k$ coins from these piles. Find the minimum number of turns you need to take away all of these coins.
Answer: 11
user: Let $M$ be a set of six distinct positive integers whose sum is $60$. These numbers are written on the faces of a cube, one number to each face. A move consists of choosing three faces of the cube that share a common vertex and adding $1$ to the numbers on those faces. Determine the number of sets $M$ for which it’s possible, after a finite number of moves, to produce a cube all of whose sides have the same number.
Answer: 84
user: Consider the pyramid $OABC$. Let the equilateral triangle $ABC$ with side length $6$ be the base. Also, $OA = OB = OC = 9$. Let $M$ be the midpoint of $AB$. Find the square of the distance from $M$ to $OC$.
Answer: 23
user: Find all prime numbers that do not have a multiple ending in $2015$.
Answer: 2
user: Determine the remainder obtained when the largest prime factor of $999999999999$ is divided by $2006$.
Answer: 1877
user: Janet played a song on her clarinet in \(3\) minutes and \(20\) seconds. Janet’s goal is to play the song at a \(25\%\) faster rate. Find the number of seconds it will take Janet to play the song when she reaches her goal.
Answer: 160
user: Tsrutsuna starts in the bottom left cell of a $7 \times 7$ square table, while Tsuna is in the upper right cell. The center cell of the table contains cheese. Tsrutsuna wants to reach Tsuna and bring a piece of cheese with him. From a cell, Tsrutsuna can only move to the right or the top neighboring cell. Determine the number of different paths Tsrutsuna can take from the lower left cell to the upper right cell, such that he passes through the center cell.
Answer: 400
user: In $\triangle ABC$, points $D$, $E$, and $F$ lie on side $\overline{BC}$ such that $\overline{AD}$ is an angle bisector of $\angle BAC$, $\overline{AE}$ is a median, and $\overline{AF}$ is an altitude. Given that $AB = 154$ and $AC = 128$, and $9 \times DE = EF$, find the side length $BC$.
Answer: 94
user: Determine all four-digit numbers $\overline{abcd}$ such that

$$a(a+b+c+d)(a^{2}+b^{2}+c^{2}+d^{2})(a^{6}+2b^{6}+3c^{6}+4d^{6})=\overline{abcd}$$
Answer: 2010
user: Let $a$, $b$, $c$, and $d$ be positive integers such that $77^a \cdot 637^b = 143^c \cdot 49^d$. Compute the minimal value of $a+b+c+d$.
Answer: 9
user: Passing through the origin of the coordinate plane are $180$ lines, including the coordinate axes, which form $1^\circ$ angles with one another at the origin. Determine the sum of the x-coordinates of the points of intersection of these lines with the line $y = 100-x$.
Answer: 8950
user: Find the sum of all 4-digit numbers using the digits \(2, 3, 4, 5, 6\) without repeating any digit.
Answer: 533280
user: On the sides of a right triangle, regular nonagons are constructed outside. These nonagons are built on one of the catheti and on the hypotenuse, with areas equal to $1602 \, \text{cm}^2$ and $2019 \, \text{cm}^2$, respectively. What is the area of the nonagon that is constructed on the other cathetus of this triangle?
Answer: 417
user: Compute $\left\lfloor\frac{1}{\frac{1}{2022}+\frac{1}{2023}+\dots+\frac{1}{2064}}\right\rfloor$.
Answer: 47
user: Let $\triangle ABC$ have $AB=3$, $AC=5$, and $\angle A=90^\circ$. Point $D$ is the foot of the altitude from $A$ to $\overline{BC}$, and $X$ and $Y$ are the feet of the altitudes from $D$ to $\overline{AB}$ and $\overline{AC}$ respectively. If $XY^2$ can be written in the form $\frac{m}{n}$ where $m$ and $n$ are positive relatively prime integers, what is $100m+n$?
Answer: 22534
user: In the diagram below, how many rectangles can be drawn using the grid lines which contain none of the letters $N$, $I$, $M$, $O$? 
\[ \text{[asy]} \]
\[
\text{size}(4\text{cm});
\text{for}(\text{int } i=0;i<6;++i)\text{draw}((i,0)--(i,5)^^(0,i)--(5,i));
\text{label}("$N$", (1.5, 2.5));
\text{label}("$I$", (2.5, 3.5));
\text{label}("$M$", (3.5, 2.5));
\text{label}("$O$", (2.5, 1.5));
\]
Answer: 73
user: Compute the number of ordered quadruples $(a, b, c, d)$ of distinct positive integers such that $\binom{\binom{a}{b}}{\binom{c}{d}} = 21$.
Answer: 13
user: Each cell of a $7 \times 7$ table is painted with one of several colors. It is known that for any two distinct rows, the numbers of colors used to paint them are distinct, and for any two distinct columns, the numbers of colors used to paint them are distinct. What is the maximum possible number of colors in the table?
Answer: 22
user: How many two-digit numbers have exactly $4$ positive factors? (Here $1$ and the number $n$ are also considered as factors of $n$.)
Answer: 30
user: If digits $A$, $B$, and $C$ (between $0$ and $9$ inclusive) satisfy:
\[
\begin{array}{c@{\,}c@{\,}c@{\,}c}
 & C & C & A \\
+ & B & 2 & B \\\hline
 & A & 8 & 8 \\
\end{array}
\]
What is $A \cdot B \cdot C$?
Answer: 42
user: The sum \[\sum_{i=0}^{1000} \frac{\binom{1000}{i}}{\binom{2019}{i}}\] can be expressed in the form $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Compute $p+q$.
Answer: 152
user: Let $\sigma(n)$ be the number of positive divisors of $n$, and let $\operatorname{rad} n$ be the product of the distinct prime divisors of $n$. By convention, $\operatorname{rad} 1 = 1$. Find the greatest integer not exceeding 
\[ 100\left(\sum_{n=1}^{\infty}\frac{\sigma(n)\sigma(n \operatorname{rad} n)}{n^2\sigma(\operatorname{rad} n)}\right)^{\frac{1}{3}}. \]
Answer: 164
user: Let $k$ be a positive integer such that $1 + \frac{1}{2} + \frac{1}{3} + \ldots + \frac{1}{13} = \frac{k}{13!}$. Find the remainder when $k$ is divided by $7$.
Answer: 1
user: Let $A=\{1,2,3,\ldots,40\}$. Find the least positive integer $k$ for which it is possible to partition $A$ into $k$ disjoint subsets with the property that if $a, b, c$ (not necessarily distinct) are in the same subset, then $a \neq b+c$.
Answer: 4
user: In $\triangle ABC$, $AB = 30$, $BC = 40$, and $CA = 50$. Squares $A_1A_2BC$, $B_1B_2AC$, and $C_1C_2AB$ are erected outside $\triangle ABC$, and the pairwise intersections of lines $A_1A_2$, $B_1B_2$, and $C_1C_2$ are $P$, $Q$, and $R$. Compute the length of the shortest altitude of $\triangle PQR$.
Answer: 124
user: Given that $P(x)$ is the least degree polynomial with rational coefficients such that 
\[P(\sqrt{2} + \sqrt{3}) = \sqrt{2},\] 
find $P(10)$.
Answer: 455
user: The function $f(n)$ is defined on the positive integers and takes non-negative integer values. Given the conditions:

- $f(2) = 0$
- $f(3) > 0$
- $f(9999) = 3333$

and for all $m, n$:

\[ f(m+n) - f(m) - f(n) = 0 \text{ or } 1. \]

Determine the value of $f(1982)$.
Answer: 660
user: Cyclic pentagon $ABCDE$ has a right angle $\angle ABC = 90^{\circ}$ and side lengths $AB = 15$ and $BC = 20$. Supposing that $AB = DE = EA$, find $CD$.
Answer: 7
user: Dima has written the numbers $\frac{1}{80!}, \frac{1}{81!}, \ldots, \frac{1}{99!}$ on 20 infinite pieces of paper as decimal fractions. For example, the following is written on the last piece: \( \frac{1}{99!} = 0.\underbrace{00\ldots00}_{155\text{ zeros}}10715\ldots \). Sasha wants to cut a fragment of $N$ consecutive digits from one of these pieces without the comma. What is the maximal $N$ for which Sasha can do this so that Dima cannot guess from which piece Sasha has cut his fragment?
Answer: 155
user: Let $ABC$ be a right triangle with a right angle at $C$. Two lines, one parallel to $AC$ and the other parallel to $BC$, intersect on the hypotenuse $AB$. These lines split the triangle into two triangles and a rectangle. The two triangles have areas $512$ and $32$. What is the area of the rectangle?
Answer: 256
user: Let \[ p(x) = x^{2008} + x^{2007} + x^{2006} + \cdots + x + 1, \] and let $r(x)$ be the polynomial remainder when $p(x)$ is divided by $x^4 + x^3 + 2x^2 + x + 1$. Find the remainder when $|r(2008)|$ is divided by $1000$.
Answer: 64
user: 2013 is the first year since the Middle Ages that consists of four consecutive digits. How many such years are there still to come after 2013 and before 10000?
Answer: 149
user: The rectangle in the figure has dimensions $16 \times 20$ and is divided into $10$ smaller equal rectangles. What is the perimeter of each of the $10$ smaller rectangles?
Answer: 24
user: We call $\overline{a_n\ldots a_2}$ the Fibonacci representation of a positive integer $k$ if 
\[k = \sum_{i=2}^n a_i F_i,\]
where $a_i \in \{0,1\}$ for all $i$, $a_n=1$, and $F_i$ denotes the $i^{\text{th}}$ Fibonacci number ($F_0=0$, $F_1=1$, and $F_i=F_{i-1}+F_{i-2}$ for all $i \ge 2$). This representation is said to be \textit{minimal} if it has fewer 1’s than any other Fibonacci representation of $k$. Find the smallest positive integer that has eight ones in its minimal Fibonacci representation.
Answer: 1596
user: Let $S$ be the set of all polynomials $Q(x,y,z)$ with coefficients in \{0,1\} such that there exists a homogeneous polynomial $P(x,y,z)$ of degree 2016 with integer coefficients and a polynomial $R(x,y,z)$ with integer coefficients so that 
\[P(x,y,z) Q(x,y,z) = P(yz,zx,xy) + 2R(x,y,z)\]
and $P(1,1,1)$ is odd. Determine the size of $S$.

Note: A homogeneous polynomial of degree $d$ consists solely of terms of degree $d$.
Answer: 509545
user: Jeff writes down the two-digit base-10 prime $\overline{ab_{10}}$. He realizes that if he misinterprets the number as the base 11 number $\overline{ab_{11}}$ or the base 12 number $\overline{ab_{12}}$, it is still a prime. What is the least possible value of Jeff’s number (in base 10)?
Answer: 61
user: We will say that a list of positive integers is *admissible* if all its numbers are less than or equal to $100$ and their sum is greater than $1810$. Find the smallest positive integer $d$ such that from each admissible list, some numbers can be crossed out such that the sum of the numbers left uncrossed is greater than or equal to $1810-d$ and less than or equal to $1810+d$.
Answer: 10
user: There are $30$ cities in the empire of Euleria. Every week, Martingale City runs a very well-known lottery. $900$ visitors decide to take a trip around the empire, visiting a different city each week in some random order. $3$ of these cities are inhabited by mathematicians, who will talk to all visitors about the laws of statistics. A visitor with this knowledge has a probability of $0$ of buying a lottery ticket, else they have a probability of $0.5$ of buying one. What is the expected number of visitors who will play the Martingale Lottery?
Answer: 405
user: There are three solutions with different percentages of alcohol. If you mix them in a ratio of $1:2:3$, you get a $20\%$ solution. If you mix them in a ratio of $5:4:3$, you will get a solution with $50\%$ alcohol content. What percentage of alcohol will the solution contain if equal amounts of the original solutions are mixed?
Answer: 40
user: A container is shaped like a square-based pyramid where the base has side length $23$ centimeters and the height is $120$ centimeters. The container is open at the base of the pyramid and stands in an open field with its vertex pointing down. One afternoon, $5$ centimeters of rain falls in the open field, partially filling the previously empty container. Find the depth in centimeters of the rainwater in the bottom of the container after the rain.
Answer: 60
user: Let $n$ be a positive integer such that $\lfloor\sqrt{n}\rfloor - 2$ divides $n - 4$ and $\lfloor\sqrt{n}\rfloor + 2$ divides $n + 4$. Find the greatest such $n$ less than $1000$. (Note: $\lfloor x \rfloor$ refers to the greatest integer less than or equal to $x$).
Answer: 956
user: Let $f_0(x) = x$, and for each $n \geq 0$, let $f_{n+1}(x) = f_n(x^2(3-2x))$. Find the smallest real number that is at least as large as \[ \sum_{n=0}^{2017} f_n(a) + \sum_{n=0}^{2017} f_n(1-a) \] for all $a \in [0,1]$.
Answer: 2018
user: A permutation $a_1, a_2, \ldots, a_{13}$ of the numbers from $1$ to $13$ is given such that $a_i > 5$ for $i=1,2,3,4,5$. Determine the maximum possible value of 
\[ a_{a_1} + a_{a_2} + a_{a_3} + a_{a_4} + a_{a_5}. \]
Answer: 45
user: There are digits $a$ and $b$ such that the 15-digit number $7a7ba7ab7ba7b77$ is divisible by 99. Find $10a + b$.
Answer: 53
user: Let $ABC$ and $DEF$ be two triangles, such that $AB = DE = 20$, $BC = EF = 13$, and $\angle A = \angle D$. If $AC - DF = 10$, determine the area of $\triangle ABC$.
Answer: 126
user: Let \([n] = \{1, 2, 3, \ldots, n\}\) and for any set \(S\), let \(P(S)\) be the set of non-empty subsets of \(S\). What is the last digit of \(|P(P([2013]))|\)?
Answer: 7
user: Let $\{a_n\}_{n=1}^{\infty}$ and $\{b_n\}_{n=1}^{\infty}$ be sequences of integers such that $a_1 = 20$, $b_1 = 15$, and for $n \ge 1$, \[\begin{aligned} a_{n+1}&=a_n^2-b_n^2, \\ b_{n+1}&=2a_nb_n-b_n^2 \end{aligned}\] Let $G = a_{10}^2-a_{10}b_{10}+b_{10}^2$. Determine the number of positive integer factors of $G$.
Answer: 525825
user: Find the smallest prime number $p$ for which the number $p^3 + 2p^2 + p$ has exactly 42 divisors.
Answer: 23