ChenDRAG/VeRL_math_validation huggingface dataset:
Problem: In preparation for the family's upcoming vacation, Tony puts together five bags of jelly beans, one bag for each day of the trip, with an equal number of jelly beans in each bag. Tony then pours all the jelly beans out of the five bags and begins making patterns with them. One of the patterns that he makes has one jelly bean in a top row, three jelly beans in the next row, five jelly beans in the row after that, and so on:

\[
\begin{array}{ccccccccc}
&&&&*&&&&\\
&&&*&*&*&&&\\
&&*&*&*&*&*&&\\
&*&*&*&*&*&*&*&\\
*&*&*&*&*&*&*&*&*\\
&&&&\vdots&&&&
\end{array}
\]

Continuing in this way, Tony finishes a row with none left over. For instance, if Tony had exactly 25 jelly beans, he could finish the fifth row above with no jelly beans left over. However, when Tony finishes, there are between 10 and 20 rows. Tony then scoops all the jelly beans and puts them all back into the five bags so that each bag once again contains the same number. How many jelly beans are in each bag? (Assume that no jelly bean gets put inside more than one bag.)
Answer: 45
Problem: Let $p$ and $q$ be positive integers such that $p$ is a prime, $p$ divides $q-1$, and $p+q$ divides $p^2+2020q^2$. Find the sum of the possible values of $p$.
Answer: 141
Problem: What is the largest integer whose prime factors add to $14$?
Answer: 162
Problem: Let $a$ be a number with $n$ digits ($n > 1$). A number $b$ with $2n$ digits is obtained by writing two copies of $a$ one after the other. If $\frac{b}{a^2}$ is an integer $k$, find the possible values of $k$.
Answer: 7
Problem: There is a pack of 27 distinct cards, and each card has three values on it. The first value is a shape from $\{\Delta,\square,\odot\}$; the second value is a letter from $\{A,B,C\}$; and the third value is a number from $\{1,2,3\}$.  

In how many ways can we choose an unordered set of 3 cards from the pack, so that no two of the chosen cards have two matching values?  

For example, we can choose $\{\Delta A1, \Delta B2, \odot C3\}$, but we cannot choose $\{\Delta A1, \square B2, \Delta C1\}$.
Answer: 1278
Problem: Find the largest $n$ for which there exists a sequence $(a_0, a_1, \ldots, a_n)$ of non-zero digits such that, for each $k$, $1 \le k \le n$, the $k$-digit number $\overline{a_{k-1} a_{k-2} \ldots a_0} = a_{k-1} 10^{k-1} + a_{k-2} 10^{k-2} + \cdots + a_0$ divides the $(k+1)$-digit number $\overline{a_{k} a_{k-1}a_{k-2} \ldots a_0}$.
Answer: 4
Problem: Compute the number of two-digit numbers $\overline{ab}$ with nonzero digits $a$ and $b$ such that $a$ and $b$ are both factors of $\overline{ab}$.
Answer: 14
Problem: Find the sum of the first $55$ terms of the sequence $\binom{0}{0}, \binom{1}{0}, \binom{1}{1}, \binom{2}{0}, \binom{2}{1}, \binom{2}{2}, \binom{3}{0}, \ldots.$ 

Note: For nonnegative integers $n$ and $k$ where $0 \leq k \leq n$, $\binom{n}{k} = \frac{n!}{k!(n-k)!}$.
Answer: 1023
Problem: Let $S$ be the set of integers of the form $2^x + 2^y + 2^z$, where $x$, $y$, $z$ are pairwise distinct non-negative integers. Determine the 100th smallest element of $S$.
Answer: 577
Problem: The alphabet in its natural order $\text{ABCDEFGHIJKLMNOPQRSTUVWXYZ}$ is $T_0$. We apply a permutation to $T_0$ to get $T_1$, which is $\text{JQOWIPANTZRCVMYEGSHUFDKBLX}$. If we apply the same permutation to $T_1$, we get $T_2$, which is $\text{ZGYKTEJMUXSODVLIAHNFPWRQCB}$. We continually apply this permutation to each $T_m$ to get $T_{m+1}$. Find the smallest positive integer $n$ such that $T_n = T_0$.
Answer: 252
Problem: Find the number of integers $n$ such that $1 \leq n \leq 2017$ and $(n-2)(n-0)(n-1)(n-7)$ is a multiple of $1001$.
Answer: 99
Problem: It is given that there exists a unique triple of positive primes \((p, q, r)\) such that \(p < q < r\) and 
\[
\frac{p^3 + q^3 + r^3}{p + q + r} = 249.
\] 
Find \(r\).
Answer: 19
Problem: On the blackboard, there are $25$ points arranged in a grid as shown in the figure below. Gastón needs to choose $4$ points that will serve as the vertices of a square. How many different ways can he make this choice?

$$
\begin{matrix}
\bullet & \bullet & \bullet & \bullet & \bullet \\
\bullet & \bullet & \bullet & \bullet & \bullet \\
\bullet & \bullet & \bullet & \bullet & \bullet \\
\bullet & \bullet & \bullet & \bullet & \bullet \\
\bullet & \bullet & \bullet & \bullet & \bullet 
\end{matrix}
$$
Answer: 50
Problem: Find the sum of the decimal digits of \[ \left\lfloor \frac{51525354555657\dots979899}{50} \right\rfloor. \] Here $\left\lfloor x \right\rfloor$ is the greatest integer not exceeding $x$.
Answer: 457
Problem: We want to paint some identically-sized cubes so that each face of each cube is painted a solid color, and each cube is painted with six different colors. If we have seven different colors to choose from, how many distinguishable cubes can we produce?
Answer: 210
Problem: How many ordered pairs of sets \((A, B)\) have the properties:

1. \( A \subseteq \{1, 2, 3, 4, 5, 6\} \)
2. \( B \subseteq \{2, 3, 4, 5, 6, 7, 8\} \)
3. \( A \cap B \) has exactly \(3\) elements.
Answer: 720
Problem: Find the number of pairs of elements, from a group of order $2011$, such that the square of the first element of the pair is equal to the cube of the second element.
Answer: 2011
Problem: Let $P$ be a point chosen on the interior of side $\overline{BC}$ of triangle $\triangle ABC$ with side lengths $\overline{AB} = 10$, $\overline{BC} = 10$, $\overline{AC} = 12$. If $X$ and $Y$ are the feet of the perpendiculars from $P$ to the sides $AB$ and $AC$, then the minimum possible value of $PX^2 + PY^2$ can be expressed as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
Answer: 2365
Problem: Let $x$ be the minimal root of the equation $x^2 - 4x + 2 = 0$. Find the first two digits after the decimal point of the number $\{x + x^2 + \cdots + x^{20}\}$, where $\{a\}$ denotes the fractional part of $a$.
Answer: 41
Problem: A polynomial $P(x)$ is a \emph{base-$n$ polynomial} if it is of the form $a_d x^d + a_{d-1} x^{d-1} + \cdots + a_1 x + a_0$, where each $a_i$ is an integer between $0$ and $n-1$ inclusive and $a_d > 0$. Find the largest positive integer $n$ such that for any real number $c$, there exists at most one base-$n$ polynomial $P(x)$ for which $P(\sqrt{2} + \sqrt{3}) = c$.
Answer: 9
Problem: Find the smallest value of the expression $$(x-1)(x-2)(x-3)(x-4) + 10$$.
Answer: 9
Problem: Find the sum of all two-digit prime numbers whose digits are also both prime numbers.
Answer: 186
Problem: There are $10$ horses, named Horse 1, Horse 2, $\ldots$, Horse 10. They get their names from how many minutes it takes them to run one lap around a circular race track: Horse $k$ runs one lap in exactly $k$ minutes. At time 0 all the horses are together at the starting point on the track. The horses start running in the same direction, and they keep running around the circular track at their constant speeds. The least time $S > 0$, in minutes, at which all $10$ horses will again simultaneously be at the starting point is $S = 2520$. Let $T>0$ be the least time, in minutes, such that at least $5$ of the horses are again at the starting point. What is the sum of the digits of $T$?

$\textbf{(A)}\ 2\ \textbf{(B)}\ 3\ \textbf{(C)}\ 4\ \textbf{(D)}\ 5\ \textbf{(E)}\ 6$
Answer: 3
Problem: Find the smallest natural number that divides $2^n + 15$ for some natural number $n$ and can be expressed in the form $3x^2 - 4xy + 3y^2$ for some integers $x$ and $y$.
Answer: 23
Problem: Determine the sum of the two largest prime factors of the integer $89! + 90!$.
Answer: 172
Problem: Two ducks, Wat and Q, are taking a math test with $1022$ other ducklings. The test consists of $30$ questions, and the $n$th question is worth $n$ points. The ducks work independently on the test. Wat answers the $n$th problem correctly with a probability of $\frac{1}{n^2}$, while Q answers the $n$th problem correctly with a probability of $\frac{1}{n+1}$. Unfortunately, the remaining ducklings each answer all $30$ questions incorrectly.

Just before turning in their test, the ducks and ducklings decide to share answers. For any question on which Wat and Q have the same answer, the ducklings change their answers to agree with them. After this process, what is the expected value of the sum of all $1024$ scores?
Answer: 1020
Problem: Find the product of all possible real values for $k$ such that the system of equations
$$x^2 + y^2 = 80$$
$$x^2 + y^2 = k + 2x - 8y$$

has exactly one real solution $(x, y)$.
Answer: 960
Problem: Let $a$, $b$, and $c$ be the 3 roots of the equation $x^3 - x + 1 = 0$. Find the value of $\frac{1}{a+1} + \frac{1}{b+1} + \frac{1}{c+1}$.
Answer: -2
Problem: In this problem, the symbol $0$ represents the number zero, the symbol $1$ represents the number seven, the symbol $2$ represents the number five, the symbol $3$ represents the number three, the symbol $4$ represents the number four, the symbol $5$ represents the number two, the symbol $6$ represents the number nine, the symbol $7$ represents the number one, the symbol $8$ represents an arbitrarily large positive integer, the symbol $9$ represents the number six, and the symbol $\infty$ represents the number eight. Compute the value of $\left|0-1+2-3^4-5+6-7^8\times9-\infty\right|$.
Answer: 90
Problem: Albert rolls a fair six-sided die thirteen times. For each roll, if the number is strictly greater than the previous roll, he gains a point. The first roll does not gain any points. Calculate the expected number of points Albert receives.
Answer: 5
Problem: On Thursday, January 1, 2015, Anna buys one book and one shelf. For the next two years, she buys one book every day and one shelf on alternate Thursdays, so she next buys a shelf on January 15th. On how many days in the period from Thursday, January 1, 2015, until (and including) Saturday, December 31, 2016, is it possible for Anna to arrange all her books on all her shelves such that there is an equal number of books on each shelf?
Answer: 89
Problem: For every integer $a > 1$, an infinite list of integers is constructed $L(a)$, as follows:

1. $a$ is the first number in the list $L(a)$.
2. Given a number $b$ in $L(a)$, the next number in the list is $b + c$, where $c$ is the largest integer that divides $b$ and is smaller than $b$.

Find all the integers $a > 1$ such that $2002$ is in the list $L(a)$.
Answer: 2002
Problem: Of the students attending a school athletic event, $80\%$ of the boys were dressed in the school colors, $60\%$ of the girls were dressed in the school colors, and $45\%$ of the students were girls. Find the percentage of students attending the event who were wearing the school colors.
Answer: 71
Problem: Reimu has a wooden cube. In each step, she creates a new polyhedron from the previous one by cutting off a pyramid from each vertex of the polyhedron. This is done along a plane through the trisection point on each adjacent edge that is closer to the vertex. For example, the polyhedron after the first step has six octagonal faces and eight equilateral triangular faces. How many faces are on the polyhedron after the fifth step?
Answer: 974
Problem: Jonathan has a magical coin machine that accepts coins in amounts of $7$, $8$, and $9$. If he puts in $7$ coins, he gets $3$ coins back; if he puts in $8$, he gets $11$ back; and if he puts in $9$, he gets $4$ back. The coin machine does not allow two entries of the same amount to happen consecutively. Starting with $15$ coins, what is the minimum number of entries he can make to end up with $4$ coins?
Answer: 4
Problem: For every positive integer $k$, let $\mathbf{T}_k = (k(k+1), 0)$, and define $\mathcal{H}_k$ as the homothety centered at $\mathbf{T}_k$ with ratio $\frac{1}{2}$ if $k$ is odd and $\frac{2}{3}$ if $k$ is even. Suppose $P = (x,y)$ is a point such that
$$(\mathcal{H}_{4} \circ \mathcal{H}_{3} \circ \mathcal{H}_2 \circ \mathcal{H}_1)(P) = (20, 20).$$
What is $x+y$?

A homothety $\mathcal{H}$ with nonzero ratio $r$ centered at a point $P$ maps each point $X$ to the point $Y$ on ray $\overrightarrow{PX}$ such that $PY = rPX$.
Answer: 256
Problem: A set of three elements is called arithmetic if one of its elements is the arithmetic mean of the other two. Likewise, a set of three elements is called harmonic if one of its elements is the harmonic mean of the other two.

How many three-element subsets of the set of integers $\{z \in \mathbb{Z} \mid -2011 < z < 2011\}$ are arithmetic and harmonic?

**Remark:** The arithmetic mean $A(a,b)$ and the harmonic mean $H(a,b)$ are defined as:

\[A(a,b) = \frac{a+b}{2} \quad \text{and} \quad H(a,b) = \frac{2ab}{a+b} = \frac{2}{\frac{1}{a} + \frac{1}{b}},\]

respectively, where $H(a,b)$ is not defined for some $a$, $b$.
Answer: 1004
Problem: Jerry and Hannah Kubik live in Jupiter Falls with their five children. Jerry works as a Renewable Energy Engineer for the Southern Company, and Hannah runs a lab at Jupiter Falls University where she researches biomass (renewable fuel) conversion rates. Michael is their oldest child, and Wendy their oldest daughter. Tony is the youngest child. Twins Joshua and Alexis are $12$ years old.

When the Kubiks went on vacation to San Diego last year, they spent a day at the San Diego Zoo. Single day passes cost $33 for adults (Jerry and Hannah), $22 for children (Michael is still young enough to get the children's rate), and family memberships (which allow the whole family in at once) cost $120. How many dollars did the family save by buying a family pass over buying single day passes for every member of the family?
Answer: 56
Problem: A positive integer is said to be "nefelibata" if, upon taking its last digit and placing it as the first digit, keeping the order of all the remaining digits intact (for example, 312 becomes 231), the resulting number is exactly double the original number. Find the smallest possible nefelibata number.
Answer: 105263157894736848
Problem: Mr. Jones teaches algebra. He has a whiteboard with a pre-drawn coordinate grid that runs from $-10$ to $10$ in both the $x$ and $y$ coordinates. Consequently, when he illustrates the graph of a quadratic, he likes to use a quadratic with the following properties:

1. The quadratic sends integers to integers.
2. The quadratic has distinct integer roots both between $-10$ and $10$, inclusive.
3. The vertex of the quadratic has integer $x$ and $y$ coordinates both between $-10$ and $10$, inclusive.

How many quadratics are there with these properties?
Answer: 478
Problem: There are $n$ sticks which have distinct integer lengths. Suppose that it's possible to form a non-degenerate triangle from any $3$ distinct sticks among them. It's also known that there are sticks of lengths $5$ and $12$ among them. What's the largest possible value of $n$ under such conditions?
Answer: 6
Problem: A natural number of five digits is called *Ecuadorian* if it satisfies the following conditions:

- All its digits are different.
- The digit on the far left is equal to the sum of the other four digits.

**Example:** 
- $91350$ is an Ecuadorian number since $9 = 1 + 3 + 5 + 0$.
- $54210$ is not since $5 \neq 4 + 2 + 1 + 0$.

**Problem:** Find how many Ecuadorian numbers exist.
Answer: 168
Problem: Find the area of quadrilateral $ABCD$ given the following conditions: two opposite angles are right angles; the two sides forming the right angle are of equal length; and the sum of the lengths of the other two sides is $10$.
Answer: 25
Problem: A four-digit number is called a "stutterer" if its first two digits are the same and its last two digits are also the same. For example, $3311$ and $2222$ are stutterer numbers. Find all stutterer numbers that are square numbers.
Answer: 7744
Problem: Let
\[ N = \sum_{a_1 = 0}^2 \sum_{a_2 = 0}^{a_1} \sum_{a_3 = 0}^{a_2} \dots \sum_{a_{2011} = 0}^{a_{2010}} \left [ \prod_{n=1}^{2011} a_n \right ]. \]

Find the remainder when $N$ is divided by 1000.
Answer: 95
Problem: Let $a_0$, $a_1$, $a_2$, $\dots$ be an infinite sequence of real numbers such that $a_0 = \frac{4}{5}$ and  
\[  
a_{n} = 2 a_{n-1}^2 - 1  
\]  
for every positive integer $n$. Let $c$ be the smallest number such that for every positive integer $n$, the product of the first $n$ terms satisfies the inequality  
\[  
a_0 a_1 \dots a_{n - 1} \le \frac{c}{2^n}.  
\]  
What is the value of $100c$, rounded to the nearest integer?
Answer: 167
Problem: For an integer $k$, let $T_k$ denote the number of $k$-tuples of integers $(x_1, x_2, \ldots, x_k)$ with $0 \le x_i < 73$ for each $i$, such that $73 \mid x_1^2 + x_2^2 + \ldots + x_k^2 - 1$. Compute the remainder when $T_1 + T_2 + \ldots + T_{2017}$ is divided by $2017$.
Answer: 2
Problem: Find the smallest three-digit divisor of the number \(1\underbrace{00\ldots 0}_{100\text{ zeros}}1\underbrace{00\ldots 0}_{100\text{ zeros}}1.\)
Answer: 111
Problem: For how many integers $x \in [0, 2007]$ is $\frac{6x^3+53x^2+61x+7}{2x^2+17x+15}$ reducible?
Answer: 286
Problem: A candy store has $100$ pieces of candy to give away. When you get to the store, there are five people in front of you, numbered from $1$ to $5$. The $i$th person in line considers the set of positive integers congruent to $i$ modulo $5$ which are at most the number of pieces of candy remaining. If this set is empty, then they take no candy. Otherwise, they pick an element of this set and take that many pieces of candy. For example, the first person in line will pick an integer from the set $\{1, 6, \dots, 96\}$ and take that many pieces of candy. How many ways can the first five people take their share of candy so that after they are done there are at least $35$ pieces of candy remaining?
Answer: 3003
Problem: A particle is moving randomly around a plane. It starts at $(0,0)$. Every second, it moves one unit randomly in a direction parallel to the $x$ or $y$ axis. At some time in the first hour, the particle was at the point $(2023,23)$. After $4092$ seconds, the particle is at $(x,y)$. Find the expected value of $x+y$.
Answer: 2046
Problem: In $\triangle{ADE}$, points $B$ and $C$ are on side $AD$, and points $F$ and $G$ are on side $AE$ such that $BG \parallel CF \parallel DE$. The area of $\triangle{ABG}$ is $36$, the area of trapezoid $CFED$ is $144$, and $AB = CD$. Find the area of trapezoid $BGFC$.
Answer: 45
Problem: The sequence $\{a_{n}\}_{n\geq 0}$ is defined by:

- $a_{0} = 20$
- $a_{1} = 100$
- $a_{n+2} = 4a_{n+1} + 5a_{n} + 20$ for $n = 0, 1, 2, \ldots$

Find the smallest positive integer $h$ satisfying $1998 \mid (a_{n+h} - a_{n})$ for all $n = 0, 1, 2, \ldots$
Answer: 108
Problem: A four-digit positive integer is called *doubly* if its first two digits form some permutation of its last two digits. For example, 1331 and 2121 are both *doubly*. How many four-digit *doubly* positive integers are there?
Answer: 171
Problem: Find $x$ such that $2^{2^{3^{2^{2}}}} = 4^{4^{x}}$.
Answer: 40
Problem: Real numbers \((x,y)\) satisfy the following equations:
\[
(x + 3)(y + 1) + y^2 = 3y
\]
\[
-x + x(y + x) = - 2x - 3.
\]
Find the sum of all possible values of \(x\).
Answer: -3
Problem: Let $P$ be a point inside the equilateral triangle $ABC$ such that $6\angle PBC = 3\angle PAC = 2\angle PCA$. Find the measure of the angle $\angle PBC$.
Answer: 15
Problem: Let $H$ be a convex, equilateral heptagon whose angles measure (in degrees) $168^\circ$, $108^\circ$, $108^\circ$, $168^\circ$, $x^\circ$, $y^\circ$, and $z^\circ$ in clockwise order. Compute the number $y$.
Answer: 132
Problem: Gwendoline rolls a pair of six-sided dice and records the product of the two values rolled. She continues rolling the dice and recording the product until one of the products appears for a third time. What is the maximum number of times Gwendoline will need to roll the two dice?
Answer: 37
Problem: A trapezium is given with parallel bases having lengths $1$ and $4$. Split it into two trapeziums by a cut, parallel to the bases, of length $3$. We now want to divide the two new trapeziums, always by means of cuts parallel to the bases, into $m$ and $n$ trapeziums, respectively, so that all the $m + n$ trapezoids obtained have the same area. Determine the minimum possible value for $m + n$ and the lengths of the cuts to be made to achieve this minimum value.
Answer: 15
Problem: Let $c_1, c_2, \ldots, c_{6030}$ be 6030 real numbers. Suppose that for any 6030 real numbers $a_1, a_2, \ldots, a_{6030}$, there exist 6030 real numbers $\{b_1, b_2, \ldots, b_{6030}\}$ such that:
\[a_n = \sum_{k=1}^{n} b_{\gcd(k,n)}\]
and
\[b_n = \sum_{d\mid n} c_d a_{n/d}\]
for $n=1, 2, \ldots, 6030$. Find $c_{6030}$.
Answer: 528
Problem: Find the sum of all positive integers whose largest proper divisor is $55$. A proper divisor of $n$ is a divisor that is strictly less than $n$.
Answer: 550
Problem: A girl and a guy are going to arrive at a train station. If they arrive within 10 minutes of each other, they will instantly fall in love and live happily ever after. But after 10 minutes, whichever one arrives first will fall asleep and they will be forever alone. The girl will arrive between 8 AM and 9 AM with equal probability. The guy will arrive between 7 AM and 8:30 AM, also with equal probability. Let $\frac{p}{q}$ for $p$, $q$ coprime be the probability that they fall in love. Find $p + q$.
Answer: 10
Problem: Let $\mathcal{A}$ denote the set of all polynomials in three variables $x, y, z$ with integer coefficients. Let $\mathcal{B}$ denote the subset of $\mathcal{A}$ formed by all polynomials which can be expressed as:
\[
(x + y + z)P(x, y, z) + (xy + yz + zx)Q(x, y, z) + xyzR(x, y, z)
\]
where $P, Q, R \in \mathcal{A}$. Find the smallest non-negative integer $n$ such that $x^i y^j z^k \in \mathcal{B}$ for all non-negative integers $i, j, k$ satisfying $i + j + k \geq n$.
Answer: 4
Problem: Two right triangles are placed next to each other to form a quadrilateral as shown. What is the perimeter of the quadrilateral?

\[ \text{[asy]} \]
\[ \text{size}(4\text{cm}); \]
\[ \text{fill}((-5,0)--(0,12)--(0,6)--(8,0)--\text{cycle}, \text{gray}+\text{opacity}(0.3)); \]
\[ \text{draw}((0,0)--(0,12)--(-5,0)--\text{cycle}); \]
\[ \text{draw}((0,0)--(8,0)--(0,6)); \]
\[ \text{label}("5", (-2.5,0), \text{S}); \]
\[ \text{label}("13", (-2.5,6), \text{dir}(140)); \]
\[ \text{label}("6", (0,3), \text{E}); \]
\[ \text{label}("8", (4,0), \text{S}); \]
\[ \text{[/asy]} \]
Answer: 42
Problem: Box is thinking of a number, whose digits are all "1". When he squares the number, the sum of its digits is $85$. How many digits does Box's number have?
Answer: 9
Problem: In rectangle $ABCD$, point $M$ is the midpoint of $AB$, and $P$ is a point on side $BC$. The perpendicular bisector of $MP$ intersects side $DA$ at point $X$. Given that $AB = 33$ and $BC = 56$, find the least possible value of $MX$.
Answer: 33
Problem: What is the remainder when \( \sum_{k=0}^{100} 10^k \) is divided by \( 9 \)?
Answer: 2
Problem: What is the largest even positive integer that cannot be expressed as the sum of two composite odd numbers?
Answer: 38
Problem: What digit must be placed in place of the "$?" in the number $888\ldots 88?999\ldots 99$ (where the $8$ and $9$ are each written $50$ times) so that the resulting number is divisible by $7$?
Answer: 5
Problem: Points $A$, $B$, $C$, and $D$ lie on a circle such that chords $\overline{AC}$ and $\overline{BD}$ intersect at a point $E$ inside the circle. Suppose that $\angle ADE = \angle CBE = 75^\circ$, $BE=4$, and $DE=8$. The value of $AB^2$ can be written in the form $a+b\sqrt{c}$ for positive integers $a$, $b$, and $c$ such that $c$ is not divisible by the square of any prime. Find $a+b+c$.
Answer: 115
Problem: In a triangle $ABC$ ($\angle BCA = 90^{\circ}$), let $D$ be the intersection of $AB$ with a circumference having a diameter $BC$. Let $F$ be the intersection of $AC$ with a line tangent to the circumference. If $\angle CAB = 46^{\circ}$, find the measure of $\angle CFD$.
Answer: 92
Problem: In a parallelogram $ABCD$, a point $P$ on the segment $AB$ is taken such that $\frac{AP}{AB} = \frac{61}{2022}$, and a point $Q$ on the segment $AD$ is taken such that $\frac{AQ}{AD} = \frac{61}{2065}$. If $PQ$ intersects $AC$ at $T$, find $\frac{AC}{AT}$ to the nearest integer.
Answer: 67
Problem: Determine all four-digit numbers $\overline{abcd}$ that are perfect squares and satisfy the equation:

$$\overline{ab} = 3 \cdot \overline{cd} + 1.$$
Answer: 2809