ChenDRAG/VeRL_math_validation huggingface dataset:
Problem: Define the sequence $a_i$ as follows: \(a_1 = 1\), \(a_2 = 2015\), and \(a_n = \frac{n a_{n-1}^2}{a_{n-1} + n a_{n-2}}\) for \(n > 2\). What is the least \(k\) such that \(a_k < a_{k-1}\)?
Answer: 7
Problem: Recall that in any row of Pascal's Triangle, the first and last elements of the row are $1$ and each other element in the row is the sum of the two elements above it from the previous row. With this in mind, define the *Pascal Squared Triangle* as follows:

- In the $n^{\text{th}}$ row, where $n \geq 1$, the first and last elements of the row equal $n^2$.
- Each other element is the sum of the two elements directly above it.

The first few rows of the Pascal Squared Triangle are shown below.

\[
\begin{array}{c@{\hspace{7em}} c@{\hspace{2pt}}c@{\hspace{2pt}}c@{\hspace{4pt}}c@{\hspace{2pt}} c@{\hspace{2pt}}c@{\hspace{2pt}}c@{\hspace{2pt}}c@{\hspace{3pt}}c@{\hspace{2pt}} c@{\hspace{2pt}}c}
\vspace{4pt}
\text{Row 1: } & & & & & & 1 & & & & & \\
\vspace{4pt}
\text{Row 2: } & & & & & 4 & & 4 & & & & \\
\vspace{4pt}
\text{Row 3: } & & & & 9 & & 8 & & 9 & & & \\
\vspace{4pt}
\text{Row 4: } & & &16& &17& &17& & 16& & \\
\vspace{4pt}
\text{Row 5: } & &25 & &33& &34 & &33 & &25 &
\end{array}
\]

Let $S_n$ denote the sum of the entries in the $n^{\text{th}}$ row. For how many integers $1 \leq n \leq 10^6$ is $S_n$ divisible by $13$?
Answer: 76926
Problem: Given a permutation $a_1, a_2, \ldots, a_{2014}$ of the numbers $1, 2, 3, \ldots, 2014$, determine the greatest number of perfect squares that can exist in the set $\{ a_1^2 + a_2, a_2^2 + a_3, a_3^2 + a_4, \ldots, a_{2013}^2 + a_{2014}, a_{2014}^2 + a_1 \}$.
Answer: 1006
Problem: Let $p$ and $q$ be prime numbers such that $(p-1)^{q-1}-1$ is a positive integer that divides $(2q)^{2p}-1$. Compute the sum of all possible values of $pq$.
Answer: 85
Problem: Three points $X$, $Y$, and $Z$ are on a straight line such that $XY = 10$ and $XZ = 3$. What is the product of all possible values of $YZ$?
Answer: 91
Problem: There are $15$ (not necessarily distinct) integers chosen uniformly at random from the range from $0$ to $999$, inclusive. Yang then computes the sum of their units digits, while Michael computes the last three digits of their sum. The probability of them getting the same result is $\frac{m}{n}$ for relatively prime positive integers $m, n$. Find $100m + n$.
Answer: 200
Problem: Determine the smallest number $N$, a multiple of 83, such that $N^2$ has 63 positive divisors.
Answer: 1992
Problem: Find the number of ending zeros of $2014!$ in base 9. Give your answer in base 9.
Answer: 616
Problem: In how many ways can one arrange seven white and five black balls in a line such that no two neighboring black balls are adjacent?
Answer: 56
Problem: A committee of 5 is to be chosen from a group of 9 people. How many ways can it be chosen, if Bill and Karl must serve together or not at all, and Alice and Jane refuse to serve with each other?
Answer: 41
Problem: In trapezoid $ABCD$ with $AD \parallel BC$, $AB = 6$, $AD = 9$, and $BD = 12$. If $\angle ABD = \angle DCB$, find the perimeter of the trapezoid.
Answer: 39
Problem: A convex quadrilateral $ABCD$ is given where $\angle DAB = \angle ABC = 45^\circ$, $DA = 3$, $AB = 7\sqrt{2}$, and $BC = 4$. Calculate the length of side $CD$.
Answer: 5
Problem: When the binomial coefficient $\binom{125}{64}$ is written out in base 10, how many zeros are at the rightmost end?
Answer: 0
Problem: $p, q, r$ are distinct prime numbers which satisfy the following equations for a natural number $A$:

$$2pqr + 50pq = 7pqr + 55pr = 8pqr + 12qr = A$$

Find all values of $A$.
Answer: 1980
Problem: The number $S$ is the result of the following sum: $1 + 10 + 19 + 28 + 37 + \ldots + 10^{2013}$. If one writes down the number $S$, how often does the digit $5$ occur in the result?
Answer: 4022
Problem: Find the number of integers $n$ with $n \ge 2$ such that the remainder when $2013$ is divided by $n$ is equal to the remainder when $n$ is divided by $3$.
Answer: 6
Problem: Find $n$ such that $4^{4^{4^2}} = 2^{8^n}$.
Answer: 11
Problem: Floyd looked at a standard $12$ hour analogue clock at $2:36$. When Floyd next looked at the clock, the angles through which the hour hand and minute hand of the clock had moved added to $247$ degrees. How many minutes after $3:00$ was that?
Answer: 14
Problem: Lynnelle and Moor love toy cars, and together, they have $27$ red cars, $27$ purple cars, and $27$ green cars. The number of red cars Lynnelle has individually is the same as the number of green cars Moor has individually. In addition, Lynnelle has $17$ more cars of any color than Moor has of any color. How many purple cars does Lynnelle have?
Answer: 22
Problem: How many ways are there to arrange the numbers $1$, $2$, $3$, $4$, $5$, $6$ on the vertices of a regular hexagon such that exactly 3 of the numbers are larger than both of their neighbors? Rotations and reflections are considered the same.
Answer: 8
Problem: The sum \( \sum_{n=0}^{2016\cdot2017^2}2018^n \) can be represented uniquely in the form \( \sum_{i=0}^{\infty}a_i\cdot2017^i \) for nonnegative integers \( a_i \) less than 2017. Compute \( a_0+a_1 \).
Answer: 1
Problem: The sum $\frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \frac{1}{6} = \frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
Answer: 69
Problem: Let $F_n$ denote the $n$th Fibonacci number, with $F_0 = 0$, $F_1 = 1$, and $F_{n} = F_{n-1} + F_{n-2}$ for $n \geq 2$. There exists a unique two-digit prime $p$ such that for all $n$, $p \mid F_{n+100} + F_n$. Find $p$.
Answer: 89
Problem: For positive integers $m$ and $n$, find the smallest possible value of $|2011^m - 45^n|$.
Answer: 14
Problem: Julius has a set of five positive integers whose mean is 100. If Julius removes the median of the set of five numbers, the mean of the remaining set increases by 5, and the median of the remaining set decreases by 5. Find the maximum possible value of the largest of the five numbers Julius has.
Answer: 269
Problem: For any positive integer $n$, we define the integer $P(n)$ by:

$$P(n) = n(n+1)(2n+1)(3n+1)\ldots(16n+1).$$

Find the greatest common divisor of the integers $P(1)$, $P(2)$, $P(3), \ldots, P(2016)$.
Answer: 510510
Problem: Let $X = \{-5,-4,-3,-2,-1,0,1,2,3,4,5\}$ and $S = \{(a,b) \in X \times X : x^2 + ax + b \text{ and } x^3 + bx + a \text{ have at least a common real zero.}\}$.

How many elements are there in $S$?
Answer: 24
Problem: Let $X$ be the sum of all divisors of the number $(3 \cdot 2009)^{((2 \cdot 2009)^{2009} - 1)}$. Find the last digit of $X$.
Answer: 0
Problem: A parliament consists of $n$ senators. These senators are organized into 10 parties and 10 committees. Each senator belongs to exactly one party and one committee. 

Determine the smallest possible value of $n$ such that it is possible to label the parties and the committees with numbers from 1 to 10, ensuring that there are at least 11 senators for which the party and committee numbers match.
Answer: 101
Problem: Let $p(x) = x^4 - 5773x^3 - 46464x^2 - 5773x + 46$. Determine the sum of $\arctan$ of its real roots.
Answer: 0
Problem: The expression $16^n + 4^n + 1$ is equivalent to the expression \( \frac{2^{p(n)} - 1}{2^{q(n)} - 1} \) for all positive integers \( n > 1 \), where \( p(n) \) and \( q(n) \) are functions and \( \frac{p(n)}{q(n)} \) is constant. Find \( p(2006) - q(2006) \).
Answer: 8024
Problem: Find the smallest integer $n$ such that any subset of $\{1, 2, \ldots, 100\}$ with $n$ elements contains at least two elements whose difference is 9.
Answer: 55
Problem: Leonhard has five cards. Each card has a nonnegative integer written on it, and any two cards show relatively prime numbers. Compute the smallest possible value of the sum of the numbers on Leonhard's cards.

Note: Two integers are relatively prime if no positive integer other than $1$ divides both numbers.
Answer: 4
Problem: For all integers $n \geq 2$, let $f(n)$ denote the largest positive integer $m$ such that $\sqrt[m]{n}$ is an integer. Evaluate 
\[f(2) + f(3) + \cdots + f(100).\]
Answer: 124
Problem: Consider an infinite sequence $x_1, x_2, \dots$ of positive integers such that, for every integer $n \geq 1$: 

- If $x_n$ is even, then $x_{n+1} = \frac{x_n}{2}$.
- If $x_n$ is odd, then $x_{n+1} = \frac{x_n - 1}{2} + 2^{k-1}$, where $2^{k-1} \leq x_n < 2^k$.

Determine the smallest possible value of $x_1$ for which $2020$ is in the sequence.
Answer: 1183
Problem: King Tin writes the first $n$ perfect squares on the royal chalkboard, but he omits the first one. For example, if $n = 3$, he writes $4$ and $9$. His son, Prince Tin, comes along and repeats the following process until only one number remains:  

He erases the two greatest numbers still on the board, calls them $a$ and $b$, and writes the value of $\frac{ab-1}{a+b-2}$ on the board.

Let $S(n)$ be the last number that Prince Tin writes on the board. Consider $\lim_{n\to \infty} S(n) = r$, meaning that $r$ is the unique number such that for every $\epsilon > 0$, there exists a positive integer $N$ so that $|S(n) - r| < \epsilon$ for all $n > N$. If $r$ can be written in simplest form as $\frac{m}{n}$, find $m + n$.
Answer: 10
Problem: Cat and Claire are having a conversation about Cat's favorite number. Cat says, "My favorite number is a two-digit positive prime integer whose first digit is less than its second, and when you reverse its digits, it's still a prime number!"

Claire asks, "If you picked a digit of your favorite number at random and revealed it to me without telling me which place it was in, is there any chance I'd know for certain what it is?"

Cat says, "Nope! However, if I now told you the units digit of my favorite number, you'd know which one it is!"

Claire says, "Now I know your favorite number!" What is Cat's favorite number?
Answer: 13
Problem: Suppose we have $10$ balls and $10$ colors. For each ball, we independently color it with one of the $10$ colors, then group the balls together by color at the end. If $S$ is the expected value of the square of the number of distinct colors used on the balls, find the sum of the digits of $S$ written as a decimal.
Answer: 55
Problem: Cat and Claire are having a conversation about Cat’s favorite number. Cat says, "My favorite number is a two-digit perfect square!"

Claire asks, "If you picked a digit of your favorite number at random and revealed it to me without telling me which place it was in, is there any chance I’d know for certain what it is?"

Cat says, "Yes! Moreover, if I told you a number and identified it as the sum of the digits of my favorite number, or if I told you a number and identified it as the positive difference of the digits of my favorite number, you wouldn’t know my favorite number."

Claire says, "Now I know your favorite number!" What is Cat’s favorite number?
Answer: 36
Problem: Compute the number of positive integers $n < 2012$ that share exactly two positive factors with 2012.
Answer: 504
Problem: Robert colors each square in an empty $3 \times 3$ grid either red or green. Find the number of colorings such that no row or column contains more than one green square.
Answer: 34
Problem: The Bell Zoo and the Carlton Zoo have various animals. The Bell Zoo has the same number of rhinoceroses as the Carlton Zoo has lions. It also has three more elephants than the Carlton Zoo has lions and the same number of elephants as the Carlton Zoo has rhinoceroses. Additionally, the Carlton Zoo has two more elephants than rhinoceroses and twice as many monkeys as it has rhinoceroses, elephants, and lions combined. It also has two more penguins than monkeys. The Bell Zoo has two-thirds as many monkeys as the Carlton Zoo has penguins, two more penguins than monkeys, and only half as many lions as penguins. The total number of rhinoceroses, elephants, lions, monkeys, and penguins in the Bell Zoo is $48$. Find the total number of rhinoceroses, elephants, lions, monkeys, and penguins in the Carlton Zoo.
Answer: 57
Problem: A function $f$ defined on the positive integers (and taking positive integer values) is given by:

$$\begin{align*}
f(1) &= 1, \\
f(3) &= 3, \\
f(2 \cdot n) &= f(n), \\
f(4 \cdot n + 1) &= 2 \cdot f(2 \cdot n + 1) - f(n), \\
f(4 \cdot n + 3) &= 3 \cdot f(2 \cdot n + 1) - 2 \cdot f(n),
\end{align*}$$

for all positive integers $n$. Determine with proof the number of positive integers $\leq 1988$ for which $f(n) = n$.
Answer: 92
Problem: Tom can run to Beth's house in $63$ minutes. Beth can run to Tom's house in $84$ minutes. At noon, Tom starts running from his house toward Beth's house while at the same time Beth starts running from her house toward Tom's house. When they meet, they both run at Beth's speed back to Beth's house. At how many minutes after noon will they arrive at Beth's house?
Answer: 72
Problem: Find the number of ways to order the integers $1, 2, 3, 4, 5, 6, 7$ from left to right so that each integer has all its divisors, besides itself, appearing to the left of it.
Answer: 150
Problem: Let $x_1 = 97$, and for $n > 1$, let $x_n = \frac{n}{x_{n - 1}}$. Calculate the product $x_1 x_2 \cdots x_8$.
Answer: 384
Problem: Let $ABCD$ be a convex quadrilateral with positive integer side lengths. Given that $\angle A = \angle B = 120^{\circ}$, $|AD - BC| = 42$, and $CD = 98$, find the maximum possible value of $AB$.
Answer: 69
Problem: Find the largest positive integer $n$ such that $n^2 + 10$ is divisible by $n-5$.
Answer: 40
Problem: Let $\triangle ABC$ be a triangle such that the area $[ABC] = 10$ and $\tan(\angle ABC) = 5$. If the smallest possible value of $(\overline{AC})^2$ can be expressed as $-a + b\sqrt{c}$ for positive integers $a, b, c$, what is $a + b + c$?
Answer: 42
Problem: Find the greatest common divisor of all numbers of the form $(2^{a^2} \cdot 19^{b^2} \cdot 53^{c^2} + 8)^{16} - 1$ where $a, b, c$ are integers.
Answer: 17
Problem: We choose 100 points in the coordinate plane. Let $N$ be the number of triples $(A, B, C)$ of distinct chosen points such that $A$ and $B$ have the same $y$-coordinate, and $B$ and $C$ have the same $x$-coordinate. Find the greatest value that $N$ can attain considering all possible ways to choose the points.
Answer: 8100
Problem: At the Lexington High School, each student is given a unique five-character ID consisting of uppercase letters. Compute the number of possible IDs that contain the string "LMT".
Answer: 2028
Problem: A mouse has a wheel of cheese which is cut into $2018$ slices. The mouse also has a $2019$-sided die, with faces labeled $0, 1, 2, \ldots, 2018$, and with each face equally likely to come up. Every second, the mouse rolls the die. If the die lands on $k$, and the mouse has at least $k$ slices of cheese remaining, then the mouse eats $k$ slices of cheese; otherwise, the mouse does nothing. What is the expected number of seconds until all the cheese is gone?
Answer: 2019
Problem: Suppose tetrahedron $PABC$ has volume $420$ and satisfies $AB = 13$, $BC = 14$, and $CA = 15$. The minimum possible surface area of $PABC$ can be written as $m+n\sqrt{k}$, where $m$, $n$, and $k$ are positive integers, and $k$ is not divisible by the square of any prime. Compute $m+n+k$.
Answer: 346
Problem: Find the remainder of $7^{1010} + 8^{2017}$ when divided by $57$.
Answer: 0
Problem: In the cyclic quadrilateral $ABXC$, it is given that $\measuredangle XAB = \measuredangle XAC$. Let $I$ be the incenter of $\triangle ABC$ and $D$ be the projection of $I$ on $\overline{BC}$. Given that $AI = 25$, $ID = 7$, and $BC = 14$, determine the length of $XI$ which can be expressed as $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers. Compute $100a + b$.
Answer: 17524
Problem: On Day $1$, Alice starts with the number $a_1=5$. For all positive integers $n>1$, on Day $n$, Alice randomly selects a positive integer $a_n$ between $a_{n-1}$ and $2a_{n-1}$, inclusive. Given that the probability that all of $a_2,a_3,\ldots,a_7$ are odd can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, compute $m+n$.
Answer: 65
Problem: Hugo, Evo, and Fidel are playing Dungeons and Dragons, which requires many twenty-sided dice. Attempting to slay Evo's vicious hobgoblin +1 of viciousness, Hugo rolls $25$ twenty-sided dice, obtaining a sum of (alas!) only $70$. Trying to console him, Fidel notes that, given that sum, the product of the numbers was as large as possible. How many $2$s did Hugo roll?
Answer: 5
Problem: You are given a convex pentagon $ABCDE$ with $AB=BC$, $CD=DE$, $\angle ABC=150^\circ$, $\angle BCD = 165^\circ$, $\angle CDE=30^\circ$, and $BD=6$. Find the area of this pentagon. Round your answer to the nearest integer if necessary.
Answer: 9
Problem: Which number do we need to subtract from the numerator and add to the denominator of $\frac{\overline{28a3}}{7276}$ such that the fraction becomes equal to $\frac{2}{7}$?
Answer: 571
Problem: Let $r_1, \dots , r_5$ be the roots of the polynomial $x^5+5x^4-79x^3+64x^2+60x+144$. What is $r_1^2+\dots+r_5^2$?
Answer: 183
Problem: Consider the sequence of numbers defined by $a_1 = 7$, $a_2 = 7^7$, and in general, $a_n = 7^{a_{n-1}}$ for $n \ge 2$. Determine the last digit of the decimal representation of $a_{2021}$.
Answer: 3
Problem: Let $\triangle ABC$ be a right triangle with legs $AB = 6$ and $AC = 8$. Let $I$ be the incenter of $\triangle ABC$ and $X$ be the other intersection of $AI$ with the circumcircle of $\triangle ABC$. Find $\overline{AI} \cdot \overline{IX}$.
Answer: 20
Problem: The cells of a $100 \times 100$ table are colored white. In one move, it is allowed to select some $99$ cells from the same row or column and recolor each of them with the opposite color. What is the smallest number of moves needed to get a table with a chessboard coloring?
Answer: 100
Problem: What is the number of ways to choose $60$ unit squares from an $11 \times 11$ chessboard such that no two chosen squares share a common side?
Answer: 62
Problem: Let $\triangle ABC$ satisfy $AB = 17$, $AC = \frac{70}{3}$, and $BC = 19$. Let $I$ be the incenter of $\triangle ABC$ and $E$ be the excenter of $\triangle ABC$ opposite $A$. (Note: this means that the circle tangent to ray $AB$ beyond $B$, ray $AC$ beyond $C$, and side $BC$ is centered at $E$.) Suppose the circle with diameter $IE$ intersects $AB$ beyond $B$ at $D$. If $BD = \frac{a}{b}$ where $a, b$ are coprime positive integers, find $a + b$.
Answer: 22
Problem: The eleven members of a cricket team are numbered $1, 2, \ldots, 11$. In how many ways can the entire cricket team sit on the eleven chairs arranged around a circular table so that the numbers of any two adjacent players differ by one or two?
Answer: 2
Problem: Richard and Shreyas are arm wrestling against each other. They will play $10$ rounds, and in each round, there is exactly one winner. If the same person wins in consecutive rounds, these rounds are considered part of the same "streak". How many possible outcomes are there in which there are strictly more than $3$ streaks? 

For example, if we denote Richard winning by $R$ and Shreyas winning by $S$, $SSRSSRRRRR$ is one such outcome, with $4$ streaks.
Answer: 932
Problem: Nine lines, parallel to the base of a triangle, divide the other two sides into 10 equal segments and the area into 10 distinct parts. Find the area of the original triangle if the area of the largest of these parts is 76.
Answer: 400
Problem: Given triangle $ABC$. Let $A_1B_1$, $A_2B_2$, $\ldots$, $A_{2008}B_{2008}$ be $2008$ lines parallel to $AB$ which divide triangle $ABC$ into $2009$ equal areas. Calculate the value of $$ \left\lfloor \frac{A_1B_1}{2A_2B_2} + \frac{A_1B_1}{2A_3B_3} + \ldots + \frac{A_1B_1}{2A_{2008}B_{2008}} \right\rfloor$$
Answer: 43
Problem: Let $S$ be the sum of all integers $b$ for which the polynomial $x^2 + bx + 2008b$ can be factored over the integers. Compute $|S|$.
Answer: 168672
Problem: Let 
$$a_k = 0.\underbrace{00\ldots 0}_{k - 1 \text{ zeros}} 1 \underbrace{00\ldots 0}_{k - 1 \text{ zeros}} 1$$
The value of \(\sum_{k = 1}^\infty a_k\) can be expressed as a rational number \(\frac{p}{q}\) in simplest form. Find \(p + q\).
Answer: 37
Problem: Caltech's 900 students are evenly spaced along the circumference of a circle. How many equilateral triangles can be formed with at least two Caltech students as vertices?
Answer: 808500
Problem: Ben "One Hunna Dolla" Franklin is flying a kite $KITE$ such that $IE$ is the perpendicular bisector of $KT$. Let $IE$ meet $KT$ at $R$. The midpoints of $KI, IT, TE, EK$ are $A, N, M, D,$ respectively. Given that $[MAKE] = 18$, $IT = 10$, $[RAIN] = 4$, find $[DIME]$.

Note: $[X]$ denotes the area of the figure $X$.
Answer: 16
Problem: Let $ABC$ be a non-equilateral triangle with integer sides. Let $D$ and $E$ be respectively the mid-points of $BC$ and $CA$; let $G$ be the centroid of $\Delta ABC$. Suppose, $D$, $C$, $E$, $G$ are concyclic. Find the least possible perimeter of $\Delta ABC$.
Answer: 37
Problem: Sandy likes to eat waffles for breakfast. To make them, she centers a circle of waffle batter with a radius of $3\text{ cm}$ at the origin of the coordinate plane. Her waffle iron imprints non-overlapping unit-square holes centered at each lattice point. How many of these holes are contained entirely within the area of the waffle?
Answer: 21
Problem: The diagram below shows square $ABCD$ with side length $12$ and square $EFGH$ with side length $6$, both having the same center. Find the area of quadrilateral $ABFE$.
Answer: 27
Problem: Determine the largest integer $n \geq 3$ for which the edges of the complete graph on $n$ vertices can be assigned pairwise distinct non-negative integers such that the edges of every triangle have numbers which form an arithmetic progression.
Answer: 4
Problem: Given that $5^{2018}$ has 1411 digits and starts with 3 (the leftmost non-zero digit is 3), for how many integers $1 \leq n \leq 2017$ does $5^n$ start with 1?
Answer: 607
Problem: The non-negative integers $x$ and $y$ satisfy the equation $\sqrt{x} + \sqrt{x + 60} = \sqrt{y}$. Determine the largest possible value of $x$.
Answer: 196
Problem: Evaluate \( \int_0^1 \frac{x+\sqrt{x}-1}{(\sqrt{x}+1)^2}\, dx \).
Answer: 0
Problem: Evariste has drawn twelve triangles such that two consecutive triangles share exactly one edge. 

Sophie colors every triangle side in red, green, or blue. Among the $3^{24}$ possible colorings, how many have the property that every triangle has one edge of each color?
Answer: 4098
Problem: Let $n$ be the product of the first $10$ primes, and let \[ S = \sum_{xy \mid n} \varphi(x) \cdot y, \] where $\varphi(x)$ denotes the number of positive integers less than or equal to $x$ that are relatively prime to $x$, and the sum is taken over ordered pairs $(x, y)$ of positive integers for which $xy$ divides $n$. Compute $\frac{S}{n}$.
Answer: 1024
Problem: Determine the sum of all possible surface areas of a cube, given that two of its vertices are $(1,2,0)$ and $(3,3,2)$.
Answer: 99
Problem: Find the two last digits of $2012^{2012}$.
Answer: 56
Problem: Paul wrote a list of all four-digit numbers such that the hundreds digit is $5$ and the tens digit is $7$. For example, $1573$ and $7570$ are on Paul's list, but $2754$ and $571$ are not. Find the sum of all the numbers on Paul's list.

$\text{Note}$. The numbers on Paul's list cannot start with zero.
Answer: 501705
Problem: The figure is created by arranging four squares, each with a side length of 5 units, on each side of a central square with a side length of 20 units. Calculate the perimeter of the entire figure.
Answer: 120
Problem: Let $S_1$, $S_2$, $\dots$, $S_{125}$ be 125 sets of 5 numbers each, comprising 625 distinct numbers. Let $m_i$ be the median of $S_i$. Let $M$ be the median of $m_1$, $m_2$, $\dots$, $m_{125}$. What is the greatest possible number of the 625 numbers that are less than $M$?
Answer: 436
Problem: Compute the number of permutations $(a,b,c,x,y,z)$ of $(1,2,3,4,5,6)$ which satisfy the five inequalities:
\[ a < b < c, \quad x < y < z, \quad a < x, \quad b < y, \quad \text{and} \quad c < z. \]
Answer: 5