ChenDRAG/VeRL_math_validation huggingface dataset:
Problem: Suppose sequence $\{a_i\} = a_1, a_2, a_3, \ldots$ satisfies $a_{n+1} = \frac{1}{a_n+1}$ for all positive integers $n$. Define $b_k$ for positive integers $k \ge 2$ to be the minimum real number such that the product $a_1 \cdot a_2 \cdot \ldots \cdot a_k$ does not exceed $b_k$ for any positive integer choice of $a_1$. Find $\frac{1}{b_2}+\frac{1}{b_3}+\frac{1}{b_4}+\ldots+\frac{1}{b_{10}}.$
Answer: 88
Problem: Let $ABC$ be a right triangle with $\angle ACB = 90^{\circ}$. $D$ is a point on $AB$ such that $CD \perp AB$. If the area of triangle $ABC$ is $84$, what is the smallest possible value of $AC^2 + (3 \cdot CD)^2 + BC^2$?
Answer: 1008
Problem: \textbf{Problem 5.} Miguel has two clocks, one clock advances $1$ minute per day and the other one goes $\frac{15}{10}$ minutes per day. If you put them at the same correct time, what is the least number of days that must pass for both to give the correct time simultaneously?
Answer: 1440
Problem: How many ways can the integers from $-7$ to $7$ inclusive be arranged in a sequence such that the absolute value of the numbers in the sequence does not decrease?
Answer: 128
Problem: Prinstan Trollner and Dukejukem are competing at the game show WASS. Both players spin a wheel which chooses an integer from $1$ to $50$ uniformly at random, and this number becomes their score. Dukejukem then flips a weighted coin that lands heads with probability $\frac{3}{5}$. If he flips heads, he adds $1$ to his score. A player wins the game if their score is higher than the other player's score. 

The probability that Dukejukem defeats Trollner to win WASS equals $\frac{m}{n}$, where $m$ and $n$ are coprime positive integers. Compute $m+n$.
Answer: 751
Problem: Find the number of $4$-digit numbers (in base $10$) that have non-zero digits and are divisible by $4$ but not by $8$.
Answer: 729
Problem: Evaluate the expression:
\[2^{7}\times 3^{0}+2^{6}\times 3^{1}+2^{5}\times 3^{2}+\cdots+2^{0}\times 3^{7}.\]
Answer: 6305
Problem: Let the sum $\sum_{n=1}^{9} \frac{1}{n(n+1)(n+2)}$ be written in its lowest terms as $\frac{p}{q}$. Find the value of $q - p$.
Answer: 83
Problem: You are using a lock with four dials, each set to a number between 0 and 9 (inclusive). You typically leave the lock with each dial one higher than the correct value. However, last night, someone changed all the values to 5. You recall that none of the digits in your code are prime numbers, 0, or 1, and the average value of the digits is 5.

**Question:**
How many combinations will you have to try?
Answer: 10
Problem: Yukihira is counting the minimum number of lines $m$ that can be drawn on the plane so that they intersect in exactly $200$ distinct points. What is $m$?
Answer: 21
Problem: Find the maximal number of crosses, each consisting of 5 squares, that can be placed on an 8x8 grid without overlapping.
Answer: 8
Problem: Compute the number of ordered pairs \((m,n)\) of positive integers that satisfy the equation \(\text{lcm}(m,n)+\gcd(m,n)=m+n+30\).
Answer: 16
Problem: Let $a = 111\ldots111$ (forty ones) and $b = 111\ldots111$ (twelve ones). Determine the greatest common divisor of $a$ and $b$.
Answer: 1111
Problem: Find the number of integers $n$ with $1 \leq n \leq 100$ for which $n - \phi(n)$ is prime. Here, $\phi(n)$ denotes the number of positive integers less than $n$ which are relatively prime to $n$.
Answer: 13
Problem: If $a_1, a_2, \ldots, a_{2013} \in [-2, 2]$ and $a_1 + a_2 + \cdots + a_{2013} = 0$, find the maximum of $a_1^3 + a_2^3 + \cdots + a_{2013}^3$.
Answer: 4026
Problem: Consider a triangle $ABC$ where $AB = AC = 15$ and $BC = 18$. Points $D$ and $E$ are chosen on $CA$ and $CB$, respectively, such that $CD = 5$ and $CE = 3$. The point $F$ is chosen on the half-line $\overrightarrow{DE}$ so that $EF = 8$. If $M$ is the midpoint of $AB$ and $N$ is the intersection of $FM$ and $BC$, what is the length of $CN$?
Answer: 9
Problem: Let $T = TNFTPP$, and let $S$ be the sum of the digits of $T$. In triangle $ABC$, points $D$, $E$, and $F$ are the feet of the angle bisectors of $\angle A$, $\angle B$, $\angle C$ respectively. Let point $P$ be the intersection of segments $AD$ and $BE$, and let $p$ denote the perimeter of $ABC$. If $AP = 3PD$, $BE = S - 1$, and $CF = 9$, then the value of $\frac{AD}{p}$ can be expressed uniquely as $\frac{\sqrt{m}}{n}$ where $m$ and $n$ are positive integers such that $m$ is not divisible by the square of any prime. Find $m + n$.
Answer: 6
Problem: Let $p(n) = n^4 - 6n^2 - 160$. If $a_n$ is the least odd prime dividing $q(n) = |p(n-30) \cdot p(n+30)|$, find $\sum_{n=1}^{2017} a_n$. ($a_n = 3$ if $q(n) = 0$.)
Answer: 7933
Problem: Compute the number of non-negative integers $k < 2^{20}$ such that $\binom{5k}{k}$ is odd.
Answer: 20736
Problem: A positive integer $n$ is called \textit{mythical} if every divisor of $n$ is two less than a prime. Find the unique mythical number with the largest number of divisors.
Answer: 135
Problem: Find the integer closest to the value of the following expression:

$$\left((3 + \sqrt{1})^{2023} - \left(\frac{1}{3 - \sqrt{1}}\right)^{2023} \right) \cdot \left((3 + \sqrt{2})^{2023} - \left(\frac{1}{3 - \sqrt{2}}\right)^{2023} \right) \cdot \ldots \cdot \left((3 + \sqrt{8})^{2023} - \left(\frac{1}{3 - \sqrt{8}}\right)^{2023} \right)$$
Answer: 0
Problem: Let $O$ and $A$ be two points in the plane with $OA = 30$, and let $\Gamma$ be a circle with center $O$ and radius $r$. Suppose that there exist two points $B$ and $C$ on $\Gamma$ with $\angle ABC = 90^{\circ}$ and $AB = BC$. Compute the minimum possible value of $\lfloor r \rfloor.$
Answer: 12
Problem: The integers from $1$ to $25$, inclusive, are randomly placed into a $5 \times 5$ grid such that in each row, the numbers are increasing from left to right. If the columns from left to right are numbered $1, 2, 3, 4,$ and $5$, then the expected column number of the entry $23$ can be written as $\frac{a}{b}$ where $a$ and $b$ are relatively prime positive integers. Find $a + b$.
Answer: 17
Problem: Consider the polynomial $x^3 - 3x^2 + 10$. Let $a$, $b$, and $c$ be its roots. Compute $a^2b^2c^2 + a^2b^2 + b^2c^2 + c^2a^2 + a^2 + b^2 + c^2$.
Answer: 169
Problem: Let $f$ be a function defined on the set of positive integers, with values in the same set, which satisfies:

- $f(n + f(n)) = 1$ for all $n \ge 1$.
- $f(1998) = 2$

Find the lowest possible value of the sum $f(1) + f(2) + \ldots + f(1999)$, and find the formula of $f$ for which this minimum is satisfied.
Answer: 1997003
Problem: Let $\Omega_1$ be a circle with center $O$, and let $AB$ be the diameter of $\Omega_1$. Let $P$ be a point on the segment $OB$, different from $O$. Suppose another circle $\Omega_2$ with center $P$ lies in the interior of $\Omega_1$. Tangents are drawn from $A$ and $B$ to the circle $\Omega_2$, intersecting $\Omega_1$ again at $A_1$ and $B_1$, respectively, such that $A_1$ and $B_1$ are on opposite sides of $AB$. Given that $A_1B = 5$, $AB_1 = 15$, and $OP = 10$, find the radius of $\Omega_1$.
Answer: 20
Problem: $x$, $y$, and $z$ are real numbers such that $xyz = 10$. What is the maximum possible value of $x^3 y^3 z^3 - 3x^4 - 12y^2 - 12z^4$?
Answer: 760
Problem: The diagram shows an $8 \times 7$ rectangle with a 3-4-5 right triangle drawn in each corner. The lower two triangles have their sides of length 4 along the bottom edge of the rectangle, while the upper two triangles have their sides of length 3 along the top edge of the rectangle. A circle is tangent to the hypotenuse of each triangle. The diameter of the circle is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
Answer: 47
Problem: Determine the last two digits of the number $Q = 2^{2017} + 2017^2$.
Answer: 61
Problem: How many real numbers $x$ are solutions to the equation $|x - 2| - 4 = \frac{1}{|x - 3|}$?
Answer: 2
Problem: There is a circle with a circumference of 1 meter and a point marked on it. Two cockroaches start running in the same direction from the marked point at different speeds. 

- Whenever the faster cockroach catches up with the slower one, the slower one instantly turns around and starts running in the opposite direction with the same speed.
- Whenever they meet face-to-face, the faster one instantly turns around and starts running in the opposite direction with the same speed.

Determine how far from the marked point their 100th meeting could occur.
Answer: 0
Problem: There are $2010$ people sitting around a round table. First, we give one person $x$ a candy. Next, we give candies to the $1^{st}$ person, $1+2^{nd}$ person, $1+2+3^{rd}$ person, $\cdots$, and $1+2+\cdots+2009^{th}$ person clockwise from $x$. Find the number of people who get at least one candy.
Answer: 408
Problem: From the digits $1, 2, \ldots, 9$, we write all the numbers formed by these nine digits (the nine digits are all distinct), and we order them in increasing order as follows: $123456789$, $123456798$, $\ldots$, $987654321$. What is the $100000^{th}$ number?
Answer: 358926471
Problem: Fill in numbers in the boxes below so that the sum of the entries in each three consecutive boxes is $2005$. What is the number that goes into the leftmost box?

```
\begin{asy}
size(300);
label("999",(2.5,.5));
label("888",(7.5,.5));

draw((0,0)--(9,0));
draw((0,1)--(9,1));
for (int i=0; i<=9; ++i)
{
draw((i,0)--(i,1));
}
\end{asy}
```

(Note: Removed extraneous information like author details and unnecessary formatting instructions.)
Answer: 118
Problem: The numbers $\frac{1}{1}, \frac{1}{2}, \ldots, \frac{1}{2010}$ are written on a blackboard. A student chooses any two of the numbers, say $x$ and $y$, erases them, and then writes down $x + y + xy$. He continues this process until only one number remains on the blackboard. What is this number?
Answer: 2010
Problem: Marko lives on the origin of the Cartesian plane. Every second, Marko moves $1$ unit up with probability $\frac{2}{9}$, $1$ unit right with probability $\frac{2}{9}$, $1$ unit up and $1$ unit right with probability $\frac{4}{9}$, and he doesn’t move with probability $\frac{1}{9}$. After $2019$ seconds, Marko ends up on the point $(A, B)$. What is the expected value of $A \cdot B$?
Answer: 1811716
Problem: A group of \( n \) friends participated in a math contest consisting of eight short-answer problems \( S_1, S_2, S_3, S_4, S_5, S_6, S_7, S_8 \) and four full-solution problems \( F_1, F_2, F_3, F_4 \). Each person in the group correctly solved exactly 11 out of the 12 problems.

We create an \( 8 \times 4 \) table. Inside the square located in the \( i \)-th row and \( j \)-th column, we write down the number of people who correctly solved both problem \( S_i \) and \( F_j \). If the 32 entries in the table sum to 256, what is the value of \( n \)?
Answer: 10
Problem: Find the greatest value of $C$ for which, for any $x, y, z, u$ such that $0 \le x \le y \le z \le u$, the inequality holds:
$$(x + y + z + u)^2 \ge C yz.$$
Answer: 9
Problem: Let $a$, $b$, and $c$ be nonzero real numbers such that $a + \frac{1}{b} = 5$, $b + \frac{1}{c} = 12$, and $c + \frac{1}{a} = 13$. Find $abc + \frac{1}{abc}$.
Answer: 750
Problem: Marisela is putting on a juggling show! She starts with $1$ ball, tossing it once per second. Lawrence tosses her another ball every five seconds, and she always tosses each ball that she has once per second. Compute the total number of tosses Marisela has made one minute after she starts juggling.
Answer: 390
Problem: Ranu starts with one standard die on a table. At each step, she rolls all the dice on the table: if all of them show a 6 on top, then she places one more die on the table; otherwise, she does nothing more on this step. After 2013 such steps, let $D$ be the number of dice on the table. What is the expected value (average value) of $6^D$?
Answer: 10071
Problem: Consider the addition problem:
\[
\begin{array}{ccccc}
 & C & A & S & H \\
+ & & & M & E \\
\hline
 O & S & I & D & E
\end{array}
\]
where each letter represents a base-ten digit, and \(C, M, O \neq 0\). Distinct letters are allowed to represent the same digit. How many ways are there to assign values to the letters so that the addition problem is true?
Answer: 0
Problem: In the obtuse triangle $ABC$ with $AB = 7$, $BC = 20$, and $CA = 15$, let point $D$ be the foot of the altitude from $C$ to line $AB$. Evaluate $[ACD] + [BCD]$. 

(Note that $[XYZ]$ denotes the area of triangle $XYZ$.)
Answer: 150
Problem: Given a positive integer $n$. The number $2n$ has $28$ positive factors, while the number $3n$ has $30$ positive factors. Find the number of positive divisors of $6n$.
Answer: 35
Problem: There exist positive integers $N$ and $M$ such that the remainders of $N$ when divided by the four integers $6$, $36$, $216$, and $M$ form an increasing nonzero geometric sequence in that order. Find the smallest possible value of $M$.
Answer: 2001
Problem: The people in an infinitely long line are numbered $1,2,3,\dots$. Then, each person says either "Karl" or "Lark" independently and at random. Let $S$ be the set of all positive integers $i$ such that people $i$, $i+1$, and $i+2$ all say "Karl," and define $X = \sum_{i \in S} 2^{-i}$. Then the expected value of $X^2$ is $\frac{m}{n}$, where $m$ and $n$ are positive integers with $\gcd(m,n)=1$. Compute $100m+n$.
Answer: 1028
Problem: Find the number of 4-digit numbers with distinct digits chosen from the set $\{0,1,2,3,4,5\}$ in which no two adjacent digits are even.
Answer: 150
Problem: Find the smallest natural number $x > 0$ such that all the following fractions are simplified:

$$\frac{3x+9}{8}, \frac{3x+10}{9}, \frac{3x+11}{10}, \ldots, \frac{3x+49}{48}$$

This means that the numerators and denominators are relatively prime.
Answer: 20
Problem: Determine all positive integers that are equal to $300$ times the sum of their digits.
Answer: 2700
Problem: Let $\mathbb{Z}_{\ge 0}$ denote the set of nonnegative integers.

Define a function $f:\mathbb{Z}_{\ge 0} \to \mathbb{Z}$ with $f\left(0\right)=1$ and 
\[ f\left(n\right)=512^{\left\lfloor n/10 \right\rfloor}f\left(\left\lfloor n/10 \right\rfloor\right) \]
for all $n \ge 1$. Determine the number of nonnegative integers $n$ such that the hexadecimal (base $16$) representation of $f\left(n\right)$ contains no more than $2500$ digits.
Answer: 10010
Problem: Find the minimum positive value of $1 \times 2 \times 3 \times \ldots \times 2022$ where you can replace $\times$ with $+$ or $-$.
Answer: 1
Problem: Find the number of pairs \((a, b)\) of positive integers such that the greatest common divisor (GCD) of \(a\) and \(b\) is equal to \(1 \cdot 2 \cdot 3 \cdot \ldots \cdot 50\), and the least common multiple (LCM) of \(a\) and \(b\) is \(1^2 \cdot 2^2 \cdot 3^2 \cdot \ldots \cdot 50^2\).
Answer: 32768
Problem: Find the least positive integer $N$ which is both a multiple of 19 and whose digits add to 23.
Answer: 779
Problem: Determine the largest integer $n$ such that $2^n$ divides the decimal representation given by some permutation of the digits $2$, $0$, $1$, and $5$. For example, $2^1$ divides $2150$. Note: The number may start with $0$.
Answer: 10
Problem: Let $\Gamma$ be the maximum possible value of $a+3b+9c$ among all triples $(a,b,c)$ of positive real numbers such that
\[ \log_{30}(a+b+c) = \log_{8}(3a) = \log_{27} (3b) = \log_{125} (3c) .\]
If $\Gamma = \frac{p}{q}$ where $p$ and $q$ are relatively prime positive integers, then find $p+q$.
Answer: 16
Problem: Mr. Vader gave out a multiple choice test, and every question had an answer that was one of A, B, or C. After the test, he curved the test so that everybody got $+50$ (so a person who got $x\%$ right would get a score of $x+50$). In the class, a score in the range $[90, \infty)$ gets an A, a score in the range $[80, 90)$ gets a B, and a score in the range $[70, 80)$ gets a C. After the curve, Mr. Vader makes this statement: "Guess A, get an A. Guess B, get a B. Guess C, get a C." That is, answering every question with the answer choice $X$ would give, with the curve, a score receiving a grade of $X$, where $X$ is one of A, B, C. Luke, a student in Mr. Vader's class, was told ahead of time that there were either $5$ or $6$ answers as A on the test. Find the sum of all possible values of the number of questions on the test, given this information.
Answer: 64
Problem: A sequence $a_0, a_1, a_2, \dots$ of real numbers satisfies $a_0 = 999$, $a_1 = -999$, and $a_n = a_{n-1}a_{n+1}$ for each positive integer $n$. Compute $\left\lvert a_1 + a_2 + \dots + a_{1000} \right\rvert$.
Answer: 1332
Problem: The \emph{Stooge sort} is a particularly inefficient recursive sorting algorithm defined as follows: given an array $A$ of size $n$, we swap the first and last elements if they are out of order. We then (if $n \ge 3$) Stooge sort the first $\lceil \frac{2n}{3} \rceil$ elements, then the last $\lceil \frac{2n}{3} \rceil$, then the first $\lceil \frac{2n}{3} \rceil$ elements again. Given that this runs in $O(n^\alpha)$, where $\alpha$ is minimal, find the value of $(243/32)^\alpha$.
Answer: 243
Problem: The diagram below shows a triangle divided into sections by three horizontal lines, which divide the altitude of the triangle into four equal parts, and three lines connecting the top vertex with points that divide the opposite side into four equal parts. If the shaded region has an area of $100$, find the area of the entire triangle.
Answer: 640
Problem: Let $M$ be a subset of the set of $2021$ integers $\{1, 2, 3, \ldots, 2021\}$ such that for any three elements (not necessarily distinct) $a, b, c$ of $M$, we have $|a + b - c| > 10$. Determine the largest possible number of elements of $M$.
Answer: 1006
Problem: For how many $x \in \{1,2,3,\dots, 2024\}$ is it possible that Bekhzod summed $2024$ non-negative consecutive integers, Ozod summed $2024+x$ non-negative consecutive integers and they got the same result?
Answer: 1138
Problem: Each number from the set $\{1, 2, 3, 4, 5, 6, 7\}$ must be written in each circle of the diagram, so that the sum of any three aligned numbers is the same (e.g., $A+D+E = D+C+B$). What number cannot be placed on the circle $E$?
Answer: 4
Problem: Point $A$, $B$, $C$, and $D$ form a rectangle in that order. Point $X$ lies on $CD$, and segments $\overline{BX}$ and $\overline{AC}$ intersect at $P$. If the area of triangle $BCP$ is 3 and the area of triangle $PXC$ is 2, what is the area of the entire rectangle?
Answer: 15
Problem: Call a positive integer $x$ a leader if there exists a positive integer $n$ such that the decimal representation of $x^n$ starts with $2012$. For example, $586$ is a leader since $586^3 = 201230056$. How many leaders are there in the set $\{1, 2, 3, \ldots, 2012\}$?
Answer: 2008
Problem: A subset $S$ of the set of integers $0$ to $99$ is said to have property $A$ if it is impossible to fill a crossword-puzzle with $2$ rows and $2$ columns using numbers in $S$ (where $0$ is written as $00$, $1$ as $01$, and so on). Determine the maximal number of elements in the set $S$ with the property $A$.
Answer: 25
Problem: Let $S$ be a set with six elements. In how many different ways can one select two not necessarily distinct subsets of $S$ so that the union of the two subsets is $S$? The order of selection does not matter; for example, the pair of subsets $\{a, c\}$, $\{b, c, d, e, f\}$ represents the same selection as the pair $\{b, c, d, e, f\}$, $\{a, c\}$.
Answer: 365
Problem: A single section at a stadium can hold either $7$ adults or $11$ children. When $N$ sections are completely filled, an equal number of adults and children will be seated in them. What is the least possible value of $N$?
Answer: 18
Problem: What is the smallest perfect square larger than $1$ with a perfect square number of positive integer factors?
Answer: 36
Problem: Define $a_k = (k^2 + 1)k!$ and $b_k = a_1 + a_2 + a_3 + \cdots + a_k$. Let \[\frac{a_{100}}{b_{100}} = \frac{m}{n}\] where $m$ and $n$ are relatively prime natural numbers. Find $n - m$.
Answer: 99
Problem: When counting from $3$ to $201$, $53$ is the $51^{\text{st}}$ number counted. When counting backwards from $201$ to $3$, $53$ is the $n^{\text{th}}$ number counted. What is $n$?

\[ \textbf{(A) } 146 \quad \textbf{(B) } 147 \quad \textbf{(C) } 148 \quad \textbf{(D) } 149 \quad \textbf{(E) } 150 \]
Answer: 149
Problem: Define the function $f(x) = \lfloor x \rfloor + \lfloor \sqrt{x} \rfloor + \lfloor \sqrt{\sqrt{x}} \rfloor$ for all positive real numbers $x$. How many integers from $1$ to $2023$ inclusive are in the range of $f(x)$? Note that $\lfloor x\rfloor$ is known as the \textit{floor} function, which returns the greatest integer less than or equal to $x$.
Answer: 1973
Problem: There are $2024$ people, consisting of knights and liars, and some of them are friends. Each person was asked how many friends they have, and the possible answers were $0, 1, \ldots, 2023$. Every knight answered truthfully, while every liar altered the real answer by exactly $1$. What is the minimal number of liars?
Answer: 1012
Problem: It is given that $2^{333}$ is a 101-digit number whose first digit is 1. How many of the numbers $2^k$, where $1 \leq k \leq 332$, have the first digit as 4?
Answer: 32
Problem: Find the period of the repetend of the fraction $\frac{39}{1428}$ by using binary numbers, i.e., its binary decimal representation.

(Note: When a proper fraction is expressed as a decimal number (of any base), either the decimal number terminates after finite steps, or it is of the form $0.b_1b_2\cdots b_sa_1a_2\cdots a_ka_1a_2\cdots a_ka_1a_2 \cdots a_k \cdots$. Here the repeated sequence $a_1a_2\cdots a_k$ is called the repetend of the fraction, and the smallest length of the repetend, $k$, is called the period of the decimal number.)
Answer: 24
Problem: Let $x$ be a real number such that:
\[ 4^{2x} + 2^{-x} + 1 = (129 + 8\sqrt{2})(4^{x} + 2^{-x} - 2^{x}). \]
Find $10x$.
Answer: 35
Problem: Define a sequence of integers $a_1, a_2, \dots, a_k$ where every term $a_i \in \{1,2\}$, and let $S$ denote their sum. Another sequence of integers $t_1, t_2, \ldots, t_k$ is defined by \[t_i=\sqrt{a_i(S-a_i)},\] for all $t_i$. Suppose that \(\sum_{1 \leq i \leq k} t_i=4000.\) Find the value of \(\sum_{1 \leq i \leq k} a^2_i.\)
Answer: 476
Problem: In triangle $ABC$, angle $A$ is twice angle $B$, angle $C$ is obtuse, and the three side lengths $a, b, c$ are integers. Determine, with proof, the minimum possible perimeter.
Answer: 77
Problem: The diagram shows a $20 \times 20$ square $ABCD$. The points $E$, $F$, and $G$ are equally spaced on side $BC$. The points $H$, $I$, $J$, and $K$ on side $DA$ are placed so that the triangles $BKE$, $EJF$, $FIG$, and $GHC$ are isosceles. Points $L$ and $M$ are midpoints of the sides $AB$ and $CD$, respectively. Find the total area of the shaded regions.
Answer: 200
Problem: It is well-known that the $n^{\text{th}}$ triangular number can be given by the formula $\frac{n(n+1)}{2}$. A Pythagorean triple of \textit{square numbers} is an ordered triple $(a,b,c)$ such that $a^2+b^2=c^2$. Let a Pythagorean triple of \textit{triangular numbers} (a PTTN) be an ordered triple of positive integers $(a,b,c)$ such that $a\leq b<c$ and
\[\frac{a(a+1)}{2}+\frac{b(b+1)}{2}=\frac{c(c+1)}{2}.\]
For instance, $(3,5,6)$ is a PTTN ($6+15=21$). Here we call both $a$ and $b$ \textit{legs} of the PTTN. Find the smallest natural number $n$ such that $n$ is a leg of \textit{at least} six distinct PTTNs.
Answer: 14
Problem: A week ago, Sandy's seasonal Little League batting average was $360$. After five more at bats this week, Sandy's batting average increased to $400$. What is the smallest number of hits that Sandy could have had this season?
Answer: 12
Problem: Let $x$ be a positive real number. Define
\[
  A = \sum_{k=0}^{\infty} \frac{x^{3k}}{(3k)!}, \quad
  B = \sum_{k=0}^{\infty} \frac{x^{3k+1}}{(3k+1)!}, \quad \text{and} \quad
  C = \sum_{k=0}^{\infty} \frac{x^{3k+2}}{(3k+2)!}.
\] 
Given that $A^3+B^3+C^3 + 8ABC = 2014$, compute $ABC$.
Answer: 183
Problem: How many ways can you fill a table of size $n \times n$ with integers such that each cell contains the total number of even numbers in its row and column, excluding itself? Two tables are considered different if they differ in at least one cell.
Answer: 1