ChenDRAG/VeRL_math_validation huggingface dataset:
Problem: We say that an integer $x \in \{1, \dots, 102\}$ is \textit{square-ish} if there exists some integer $n$ such that $x \equiv n^2 + n \pmod{103}$. Compute the product of all \textit{square-ish} integers modulo $103$.
Answer: 52
Problem: A palindrome number is a positive integer that reads the same forward and backward. For example, $1221$ and $8$ are palindrome numbers, whereas $69$ and $157$ are not. 

Let $A$ and $B$ be $4$-digit palindrome numbers, and $C$ be a $3$-digit palindrome number. Given that $A - B = C$, what is the value of $C$?
Answer: 121
Problem: Let $m = 30030$ and let $M$ be the set of its positive divisors which have exactly $2$ prime factors. Determine the smallest positive integer $n$ with the following property: for any choice of $n$ numbers from $M$, there exist 3 numbers $a$, $b$, $c$ among them satisfying $abc = m$.
Answer: 11
Problem: The Purple Comet! Math Meet runs from April 27 through May 3, so the sum of the calendar dates for these seven days is $27 + 28 + 29 + 30 + 1 + 2 + 3 = 120$. What is the largest sum of the calendar dates for seven consecutive Fridays occurring at any time in any year?
Answer: 142
Problem: How many ways are there to create two 3-digit numbers $m$ and $n$ such that $n = 3m$? Each of the digits $1$, $2$, $3$, $6$, $7$, and $8$ must be used exactly once.
Answer: 2
Problem: What is the integer closest to $\pi^{\pi}$?
Answer: 36
Problem: Let $a$ and $b$ be positive real numbers such that $\log_{43}{a} = \log_{47} (3a + 4b) = \log_{2021}b^2$. Then, the value of $\frac{b^2}{a^2}$ can be written as $m + \sqrt{n}$, where $m$ and $n$ are integers. Find $m + n$.
Answer: 123
Problem: A square has vertices $(0,10)$, $(0, 0)$, $(10, 0)$, and $(10,10)$ on the $x-y$ coordinate plane. A second quadrilateral is constructed with vertices $(0,10)$, $(0, 0)$, $(10, 0)$, and $(15,15)$. Find the positive difference between the areas of the original square and the second quadrilateral.
Answer: 50
Problem: The largest prime factor of $199^4 + 4$ has four digits. Compute the second largest prime factor.
Answer: 181
Problem: How many permutations $p(n)$ of the set $\{1, 2, \dots, 35\}$ satisfy the condition that if $a \mid b$ (where $a$ divides $b$), then $p(a) \mid p(b)$?
Answer: 48
Problem: How many ways can you arrange $3$ Alice's, $1$ Bob, $3$ Chad's, and $1$ David in a line if the Alice's are all indistinguishable, the Chad's are all indistinguishable, and Bob and David want to be adjacent to each other? In other words, how many ways can you arrange $3$ A's, $1$ B, $3$ C's, and $1$ D in a row where the B and D are adjacent?
Answer: 280
Problem: Let $a, b, c, d$ be real numbers satisfying the system of equations:
\[
\begin{align*}
(a+b)(c+d) &= 2, \\
(a+c)(b+d) &= 3, \\
(a+d)(b+c) &= 4.
\end{align*}
\]
Find the minimum value of $a^2 + b^2 + c^2 + d^2$.
Answer: 7
Problem: Let $ABC$ be a triangle with $AB = 5$, $AC = 8$, and $BC = 7$. Let $D$ be on side $AC$ such that $AD = 5$ and $CD = 3$. Let $I$ be the incenter of triangle $ABC$ and $E$ be the intersection of the perpendicular bisectors of $\overline{ID}$ and $\overline{BC}$. Suppose $DE = \frac{a\sqrt{b}}{c}$ where $a$ and $c$ are relatively prime positive integers, and $b$ is a positive integer not divisible by the square of any prime. Find $a+b+c$.
Answer: 13
Problem: There are $n$ players in a round-robin ping-pong tournament, meaning every two persons will play exactly one game. After some matches have been played, it is known that the total number of matches that have been played among any $n-2$ people is equal to $3^k$, where $k$ is a fixed integer. Find the sum of all possible values of $n$.
Answer: 9
Problem: Find all four-digit natural numbers $\overline{xyzw}$ such that the sum of the number and the sum of its digits equals $2003$.
Answer: 1978
Problem: Let $M$ be the set of all integers from $1$ to $2013$. Each subset of $M$ is assigned one of $k$ available colors. The only condition is that if the union of two different subsets $A$ and $B$ equals $M$, then $A$ and $B$ must be given different colors. What is the least possible value of $k$?
Answer: 2014
Problem: Let $\mathcal{S}$ be the set of all positive integers which are both a multiple of $3$ and have at least one digit that is a $1$. For example, $123$ is in $\mathcal{S}$ and $450$ is not. The probability that a randomly chosen $3$-digit positive integer is in $\mathcal{S}$ can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
Answer: 197
Problem: Caroline wants to plant 10 trees in her orchard. Planting $n$ apple trees requires $n^2$ square meters, planting $n$ apricot trees requires $5n$ square meters, and planting $n$ plum trees requires $n^3$ square meters. If she is committed to growing only apple, apricot, and plum trees, what is the least amount of space in square meters that her garden will take up?
Answer: 40
Problem: Let $ABC$ be a triangle with angles $A$, $B$, and $C$. Suppose $\tan A$, $\tan B$, $\tan C$ form a geometric sequence in that order. If $1 \le \tan A + \tan B + \tan C \le 2015$, find the number of possible integer values for $\tan B$.
Answer: 11
Problem: Pål has more chickens than he can manage to keep track of. Therefore, he keeps an index card for each chicken. He keeps the cards in ten boxes, each of which has room for $2021$ cards.

Unfortunately, Pål is quite disorganized, so he may lose some of his boxes. Therefore, he makes several copies of each card and distributes them among different boxes, so that even if he can only find seven boxes, no matter which seven, these seven boxes taken together will contain at least one card for each of his chickens.

What is the largest number of chickens Pål can keep track of using this system?
Answer: 5052
Problem: In the parallelogram $ABCD$, we are given that $AC = 10$ and $BD = 28$. The points $K$ and $L$ are such that they move in the plane of $ABCD$ satisfying $AK = BD$ and $BL = AC$. Let $M$ and $N$ be the midpoints of $CK$ and $DL$, respectively. Determine the maximum value of the expression $\cot^2 \left(\frac{\angle BMD}{2}\right) + \tan^2 \left(\frac{\angle ANC}{2}\right)$.
Answer: 2
Problem: A circle is inscribed in a regular octagon with area $2024$. A second regular octagon is inscribed in the circle, and its area can be expressed as $a + b\sqrt{c}$, where $a$, $b$, $c$ are integers and $c$ is square-free. Compute $a + b + c$.
Answer: 1520
Problem: Let \( a\uparrow\uparrow b = a^{a^{\cdots^{a}}} \), where there are \( b \) instances of \( a \). The operation \( a\uparrow\uparrow b \) is defined recursively as follows:  
\[  a\uparrow\uparrow b = \begin{cases} a & \text{if } b=1 \\ a^{a\uparrow\uparrow (b-1)} & \text{if } b\ge2 \end{cases} \]  
Determine the remainder of \( 3\uparrow\uparrow( 3\uparrow\uparrow ( 3\uparrow\uparrow 3)) \) when divided by \( 60 \).
Answer: 27
Problem: Call a date \( mm/dd/yy \) \textit{multiplicative} if its month number times its day number is a two-digit integer equal to its year expressed as a two-digit year. For example, \( 01/21/21 \), \( 03/07/21 \), and \( 07/03/21 \) are multiplicative. Find the number of dates between January 1, 2022 and December 31, 2030 that are multiplicative.
Answer: 29
Problem: In the plane, there are $2020$ points, some of which are black and the rest are green. For every black point, the following condition holds: there are exactly two green points that are at a distance of $2020$ from that black point. Find the smallest possible number of green dots.
Answer: 45
Problem: Farmer John has $5$ cows, $4$ pigs, and $7$ horses. How many ways can he pair up the animals so that every pair consists of animals of different species? Assume that all animals are distinguishable from each other.
Answer: 100800
Problem: Eva draws an equilateral triangle and its altitudes. In the first step, she draws the center triangle of the equilateral triangle. In the second step, she draws the center triangle of this center triangle, and so on. After each step, Eva counts all triangles whose sides lie completely on the drawn lines. What is the minimum number of center triangles she must have drawn so that the figure contains more than 2022 such triangles?
Answer: 7
Problem: There are $24$ participants who attended a meeting. Each pair of them either shook hands once or did not. A total of $216$ handshakes occurred during the meeting. For any two participants who have shaken hands, at most $10$ among the remaining $22$ participants have shaken hands with exactly one of these two persons. Define a *friend circle* as a group of $3$ participants where each person has shaken hands with the other two. Find the minimum possible number of friend circles.
Answer: 864
Problem: Find the value of $2(\sin 2^{\circ} \tan 1^{\circ} + \sin 4^{\circ} \tan 1^{\circ} + \cdots + \sin 178^{\circ} \tan 1^{\circ})$.
Answer: 2
Problem: A bag contains $8$ green candies and $4$ red candies. You randomly select one candy at a time to eat. If you eat five candies, there are relatively prime positive integers $m$ and $n$ such that $\frac{m}{n}$ is the probability that you do not eat a green candy after you eat a red candy. Find $m+n$.
Answer: 6
Problem: A sequence with the first term $a_0$ is defined such that $a_{n+1} = 2a_n^2 - 1$ for $n \geq 0$. Let $N$ denote the number of possible values of $a_0$ such that $a_0 = a_{2020}$. Find the number of factors of $N$.
Answer: 4
Problem: Ben flips a coin 10 times and records the absolute difference between the total number of heads and tails. He then flips the coin one more time and records the absolute difference again. If the probability that the second recorded number is greater than the first can be expressed as \( \frac{a}{b} \) for positive integers \( a, b \) with \( \gcd(a, b) = 1 \), find \( a + b \).
Answer: 831
Problem: Find the number of positive integers less than $2018$ that are divisible by $6$ but are not divisible by at least one of the numbers $4$ or $9$.
Answer: 280
Problem: In how many ways can you divide the set of eight numbers $\{2, 3, \ldots, 9\}$ into $4$ pairs such that no pair of numbers has a greatest common divisor (gcd) equal to $2$?
Answer: 36
Problem: Find $x$ such that 
\[2010^{\log_{10}x} = 11^{\log_{10}(1+3+5+\cdots +4019)}.\]
Answer: 121
Problem: Find the number of positive $6$-digit integers such that the sum of their digits is $9$, and four of its digits are $2, 0, 0, 4$.
Answer: 180
Problem: Let $n$ be a natural number. A cube of edge $n$ may be divided into 1996 smaller cubes, each having edge lengths that are also natural numbers. Find the minimum possible value for $n$.
Answer: 13
Problem: Let \((x_n)\) be a sequence of positive integers defined as follows: \(x_1\) is a fixed six-digit number and for any \(n \geq 1\), \(x_{n+1}\) is a prime divisor of \(x_n + 1\). Find \(x_{19} + x_{20}\).
Answer: 5
Problem: Find the sum of the real roots of the polynomial:
\[
\prod_{k=1}^{100} \left( x^2-11x+k \right) = \left( x^2-11x+1 \right)\left( x^2-11x+2 \right)\dots\left(x^2-11x+100\right).
\]
Answer: 330
Problem: Compute how many permutations of the numbers $1, 2, \dots, 8$ have no adjacent numbers that sum to $9$.
Answer: 13824
Problem: A nonempty word is called pronounceable if it alternates between vowels (A, E, I, O, U) and consonants (all other letters) and contains at least one vowel. How many pronounceable words can be formed using the letters P, U, M, A, C at most once each? Words of length shorter than $5$ are allowed.
Answer: 68
Problem: Let $B$, $M$, and $T$ be the three roots of the equation $x^3 + 20x^2 - 18x - 19 = 0$. What is the value of $|(B + 1)(M + 1)(T + 1)|$?
Answer: 18
Problem: What is the smallest perfect square that ends in \( 9009 \)?
Answer: 1503
Problem: On the number line, consider the point $x$ that corresponds to the value $10$. Consider $24$ distinct integer points $y_1, y_2, \ldots, y_{24}$ on the number line such that for all $k$ where $1 \leq k \leq 12$, we have $y_{2k-1}$ is the reflection of $y_{2k}$ across $x$. Find the minimum possible value of \[\sum_{n=1}^{24} (|y_n-1| + |y_n+1|).\]
Answer: 494
Problem: William is biking from his home to his school and back, using the same route. When he travels to school, there is an initial $20^\circ$ incline for $0.5$ kilometers, a flat area for $2$ kilometers, and a $20^\circ$ decline for $1$ kilometer. If William travels at $8$ kilometers per hour during uphill $20^\circ$ sections, $16$ kilometers per hour during flat sections, and $20$ kilometers per hour during downhill $20^\circ$ sections, find the closest integer to the number of minutes it takes William to get to school and back.
Answer: 31
Problem: How many integers $n$ are there such that $(n+1!)(n+2!)(n+3!)\cdots(n+2013!)$ is divisible by $210$ and $1 \le n \le 210$?
Answer: 120
Problem: At the beginning of a game, the numbers $1, 2, \ldots, 2004$ are written on a desk. A move consists of:

1. Selecting some numbers from those on the desk.
2. Calculating the remainder of the sum of these numbers when divided by $11$ and writing this remainder on the desk.
3. Deleting the selected numbers.

In this game, after several moves, only two numbers remained on the desk. One of them was $1000$. What was the other number?
Answer: 4
Problem: Triangle $ABC$ has side lengths $AB = 13$, $AC = 14$, and $BC = 15$ and centroid $G$. What is the area of the triangle with side lengths $AG$, $BG$, and $CG$?
Answer: 28
Problem: 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
Problem: Find the number of positive integer divisors of $12!$ that leave a remainder of $1$ when divided by $3$.
Answer: 66
Problem: Find the smallest positive integer solution to the equation $2^{2^k} \equiv k \pmod{29}$.
Answer: 24
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: Let $n$ be a positive integer. What is the smallest sum of digits that $5^n + 6^n + 2022^n$ can take?
Answer: 8
Problem: 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
Problem: 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
Problem: 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
Problem: Compute the sum of the distinct prime factors of $10101$.
Answer: 60
Problem: 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
Problem: 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
Problem: 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
Problem: What is the smallest positive integer $n$ such that $20 \equiv n^{15} \pmod{29}$?
Answer: 20
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: Find the circumradius of the triangle with side lengths $104$, $112$, and $120$.
Answer: 65
Problem: 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
Problem: 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
Problem: 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
Problem: How many positive integers between 1 and 400 (inclusive) have exactly 15 positive integer factors?
Answer: 3
Problem: 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
Problem: Calculate the value of $\sqrt{\frac{11^4+100^4+111^4}{2}}$ and provide your answer as an integer.
Answer: 11221
Problem: Find the number of ordered pairs of integers \((a, b)\) that satisfy the inequality

\[
1 < a < b+2 < 10.
\]
Answer: 28
Problem: 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
Problem: $q$ 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: \( \triangle ABC \) 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
Problem: 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
Problem: 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