ChenDRAG/VeRL_math_validation huggingface dataset:
Problem: Let $a$, $b$, $c$, and $d$ be prime numbers such that $a > 3b > 6c > 12d$ and $a^2 - b^2 + c^2 - d^2 = 1749$. Determine all possible values of $a^2 + b^2 + c^2 + d^2$.
Answer: 1999
Problem: Ethan Song and Bryan Guo are playing an unfair game of rock-paper-scissors. In any game, Ethan has a \( \frac{2}{5} \) chance to win, \( \frac{2}{5} \) chance to tie, and \( \frac{1}{5} \) chance to lose. How many games is Ethan expected to win before losing?
Answer: 2
Problem: Let $ABC$ be a triangle with $BC = 7$, $AB = 5$, and $AC = 8$. Let $M$ and $N$ be the midpoints of sides $AC$ and $AB$, respectively, and let $O$ be the circumcenter of $\triangle ABC$. Let $BO$ and $CO$ meet $AC$ and $AB$ at $P$ and $Q$, respectively. If $MN$ meets $PQ$ at $R$ and $OR$ meets $BC$ at $S$, then the value of $OS^2$ can be written in the form $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $100m + n$.
Answer: 240607
Problem: Dragoons take up $1 \times 1$ squares in the plane with sides parallel to the coordinate axes such that the interiors of the squares do not intersect. A dragoon can fire at another dragoon if the difference in the $x$-coordinates of their centers and the difference in the $y$-coordinates of their centers are both at most $6$, regardless of any dragoons in between. For example, a dragoon centered at $(4, 5)$ can fire at a dragoon centered at the origin, but a dragoon centered at $(7, 0)$ cannot. A dragoon cannot fire at itself. What is the maximum number of dragoons that can fire at a single dragoon simultaneously?
Answer: 168
Problem: Let $\pi$ be a permutation of the numbers from $1$ through $2012$. What is the maximum possible number of integers $n$ with $1 \le n \le 2011$ such that $\pi(n)$ divides $\pi(n + 1)$?
Answer: 1006
Problem: Consider the graph in 3-space of
\[0 = xyz(x + y)(y + z)(z + x)(x - y)(y - z)(z - x).\]
This graph divides 3-space into \(N\) connected regions. What is \(N\)?
Answer: 48
Problem: The integers $1$ through $1000$ are arranged on the circumference of a circle in natural order. Starting with $1$, every fifteenth number (i.e., $1, 16, 31, \ldots$) is marked. The marking continues until an already marked number is reached. How many numbers will remain unmarked?
Answer: 800
Problem: Find the largest positive integer $k$ for which there exists a convex polyhedron $\mathcal{P}$ with 2022 edges, which satisfies the following properties:

- The degrees of the vertices of $\mathcal{P}$ don’t differ by more than one, and
- It is possible to color the edges of $\mathcal{P}$ with $k$ colors such that for every color $c$, and every pair of vertices $(v_1, v_2)$ of $\mathcal{P}$, there is a monochromatic path between $v_1$ and $v_2$ in the color $c$.
Answer: 2
Problem: Compute the number of positive integers $n$ satisfying the inequalities:
\[
2^{n-1} < 5^{n-3} < 3^n.
\]
Answer: 5
Problem: Vasya has $n$ candies of several types, where $n > 145$. It is known that for any group of at least 145 candies, there is a type of candy which appears exactly 10 times. Find the largest possible value of $n$.
Answer: 160
Problem: Call a positive integer $n \geq 2$ "junk" if there exist two distinct $n$-digit binary strings $a_1a_2\cdots a_n$ and $b_1b_2\cdots b_n$ such that:

- $a_1 + a_2 = b_1 + b_2$,
- $a_{i-1} + a_i + a_{i+1} = b_{i-1} + b_i + b_{i+1}$ for all $2 \leq i \leq n-1$,
- $a_{n-1} + a_n = b_{n-1} + b_n$.

Find the number of junk positive integers less than or equal to $2016$.
Answer: 672
Problem: Let $p$ be a constant number such that $0 < p < 1$. Evaluate

\[
\sum_{k=0}^{2004} \frac{p^k (1-p)^{2004-k}}{\int_0^1 x^k (1-x)^{2004-k} \, dx}
\]
Answer: 2005
Problem: The three roots of the cubic $30x^3 - 50x^2 + 22x - 1$ are distinct real numbers between $0$ and $1$. For every nonnegative integer $n$, let $s_n$ be the sum of the $n$th powers of these three roots. What is the value of the infinite series \[ s_0 + s_1 + s_2 + s_3 + \dots \] ?
Answer: 12
Problem: Let $ABCD$ be a parallelogram with an area of 160. The diagonals $AC$ and $BD$ intersect at point $E$. Point $P$ is on the line segment $\overline{AE}$ such that $EC = 4EP$. If the line $DP$ intersects $AB$ at point $F$, find the area of quadrilateral $BFPC$.
Answer: 62
Problem: If $\phi$ is the Golden Ratio, we know that $\frac{1}{\phi} = \phi - 1$. Define a new positive real number, called $\phi_d$, where $\frac{1}{\phi_d} = \phi_d - d$ (so $\phi = \phi_1$). Given that $\phi_{2009} = \frac{a + \sqrt{b}}{c}$, where $a$, $b$, $c$ are positive integers, and the greatest common divisor of $a$ and $c$ is 1, find $a + b + c$.
Answer: 4038096
Problem: In $\triangle ABC$, $AB = AC$. Its circumcircle, $\Gamma$, has a radius of 2. Circle $\Omega$ has a radius of 1 and is tangent to $\Gamma$, $\overline{AB}$, and $\overline{AC}$. The area of $\triangle ABC$ can be expressed as $\frac{a\sqrt{b}}{c}$ for positive integers $a, b, c$, where $b$ is squarefree and $\gcd (a, c) = 1$. Compute $a + b + c$.
Answer: 339
Problem: Zang is at the point $(3,3)$ in the coordinate plane. Every second, he can move one unit up or one unit right, but he may never visit points where the $x$ and $y$ coordinates are both composite. In how many ways can he reach the point $(20, 13)$?
Answer: 210
Problem: Find the sum of all positive integers $n$ with the following properties:

- $n$ is not divisible by any primes larger than $10$.
- For some positive integer $k$, the positive divisors of $n$ are ordered as:
  \[1 = d_1 < d_2 < d_3 < \cdots < d_{2k} = n.\]
- The divisors of $n$ satisfy the condition:
  \[d_1 + d_2 + \cdots + d_k = 3k.\]
Answer: 416
Problem: Let $\clubsuit(x)$ denote the sum of the digits of the positive integer $x$. For example, $\clubsuit(8) = 8$ and $\clubsuit(123) = 1 + 2 + 3 = 6$. For how many two-digit values of $x$ is $\clubsuit(\clubsuit(x)) = 3$?
Answer: 10
Problem: Let $a$ and $b$ be complex numbers such that $(a+1)(b+1)=2$ and $(a^2+1)(b^2+1)=32.$ Compute the sum of all possible values of $(a^4+1)(b^4+1).$
Answer: 1160
Problem: Let $S$ be the set of all subsets of $\{2,3,\ldots,2016\}$ with size $1007$, and for a nonempty set $T$ of numbers, let $f(T)$ be the product of the elements in $T$. Determine the remainder when \[ \sum_{T\in S}\left(f(T)-f(T)^{-1}\right)^2\] is divided by $2017$. Note: For $b$ relatively prime to $2017$, $b^{-1}$ is the unique positive integer less than $2017$ for which $2017$ divides $bb^{-1} -1$.
Answer: 2014
Problem: Find the smallest positive prime that divides $n^2 + 5n + 23$ for some integer $n$.
Answer: 17
Problem: Farmer John has a flexible fence of length $L$ and two straight walls that intersect at a corner perpendicular to each other. He knows that if he doesn't use any walls, he can enclose a maximum possible area of $A_0$. When he uses one of the walls or both walls, he gets a maximum area of $A_1$ and $A_2$, respectively. If $n=\frac{A_1}{A_0}+\frac{A_2}{A_1}$, find $\lfloor 1000n \rfloor$.
Answer: 4000
Problem: Let $\{a_n\}_{n \geq 1}$ be an increasing and bounded sequence. Calculate the limit:

$$\lim_{{n \to \infty}} \left( 2a_n - a_1 - a_2 \right) \left( 2a_n - a_2 - a_3 \right) \cdots \left( 2a_n - a_{n-2} - a_{n-1} \right) \left( 2a_n - a_{n-1} - a_1 \right).$$
Answer: 0
Problem: The positive integers $a, b, c$ satisfy the following conditions:

1. $\text{gcd}(a, b, c) = 1$
2. $\text{gcd}(a, b + c) > 1$
3. $\text{gcd}(b, c + a) > 1$
4. $\text{gcd}(c, a + b) > 1$

Determine the smallest possible value of $a + b + c$.
Answer: 30
Problem: Let $m$ and $n$ be positive integers such that $\gcd(m,n)=1$ and $$\sum_{k=0}^{2020} (-1)^k {2020 \choose k}  \cos(2020\cos^{-1}(\tfrac{k}{2020}))=\frac{m}{n}.$$ Suppose $n$ is written as the product of a collection of (not necessarily distinct) prime numbers. Compute the sum of the members of this collection. (For example, if it were true that $n=12=2\times 2\times 3$, then the answer would be $2+2+3=7$.)
Answer: 209601
Problem: Triangle $ABC$ has sides $AB = 4$, $AC = 5$, and $BC = 6$. An angle bisector is drawn from angle $A$ and meets $BC$ at point $M$. Find the nearest integer to $100 \frac{AM}{CM}$.
Answer: 100
Problem: Farmer James invents a new currency such that for every positive integer $n \le 6$, there exists an $n$-coin worth $n!$ cents. Furthermore, he has exactly $n$ copies of each $n$-coin. An integer $k$ is said to be \textit{nice} if Farmer James can make $k$ cents using at least one copy of each type of coin. How many positive integers less than 2018 are nice?
Answer: 210
Problem: Let $L_n$ be the least common multiple of the integers $1, 2, \dots, n$. For example, $L_{10} = 2,520$ and $L_{30} = 2,329,089,562,800$. Find the remainder when $L_{31}$ is divided by $100,000$.
Answer: 46800
Problem: Find all three-digit numbers such that the square of the number is equal to the sum of their digits raised to the power of 5.
Answer: 243
Problem: Let $ABC$ be an equilateral triangle with side length $10$. A square $PQRS$ is inscribed in it, with $P$ on $AB$, $Q$ and $R$ on $BC$, and $S$ on $AC$. If the area of the square $PQRS$ is $m + n\sqrt{k}$ where $m$ and $n$ are integers and $k$ is a prime number, then determine the value of $\sqrt{\frac{m+n}{k^2}}$.
Answer: 10
Problem: A kite is inscribed in a circle with center $O$ and radius $60$. The diagonals of the kite meet at a point $P$, and $OP$ is an integer. The minimum possible area of the kite can be expressed in the form $a\sqrt{b}$, where $a$ and $b$ are positive integers and $b$ is squarefree. Find $a+b$.
Answer: 239
Problem: Let $A$, $B$, and $C$ be unique collinear points such that $AB = BC = \frac{1}{3}$. Let $P$ be a point that lies on the circle centered at $B$ with radius $\frac{1}{3}$ and the circle centered at $C$ with radius $\frac{1}{3}$. Find the measure of angle $\angle PAC$ in degrees.
Answer: 30
Problem: Determine the largest possible number of groups one can compose from the integers $1, 2, 3, \ldots, 19, 20$, so that the product of the numbers in each group is a perfect square. Each group may contain exactly one number, in which case the product equals this number, and each number must be in exactly one group.
Answer: 8
Problem: 56 lines are drawn on a plane such that no three of them are concurrent. If the lines intersect at exactly 594 points, what is the maximum number of them that could have the same slope?
Answer: 44
Problem: Let $n$ be a positive integer such that $12n^2 + 12n + 11$ is a four-digit number with all four digits equal. Determine the value of $n$.
Answer: 21
Problem: Suppose there are $160$ pigeons and $n$ holes. The $1$st pigeon flies to the $1$st hole, the $2$nd pigeon flies to the $4$th hole, and so on, such that the $i$th pigeon flies to the $(i^2 \mod n)$th hole, where $k \mod n$ is the remainder when $k$ is divided by $n$. What is the minimum $n$ such that there is at most one pigeon per hole?
Answer: 326
Problem: Find the maximum value of an integer $B$ such that for every 9 distinct natural numbers with the sum of $2023$, there must exist a sum of 4 of these numbers that is greater than or equal to $B$.
Answer: 910
Problem: For any positive integer $n$, let \( a_n = \sum_{k=1}^{\infty} \left\lfloor \frac{n+2^{k-1}}{2^k} \right\rfloor \), where \( \left\lfloor x \right\rfloor \) is the largest integer that is equal to or less than \( x \). Determine the value of \( a_{2015} \).
Answer: 2015
Problem: Let $a$, $b$, $c$ be positive reals for which
\[
  (a+b)(a+c) = bc + 2 \\
  (b+c)(b+a) = ca + 5 \\
  (c+a)(c+b) = ab + 9
\]
If $abc = \frac{m}{n}$ for relatively prime positive integers $m$ and $n$, compute $100m+n$.
Answer: 4532
Problem: Let $ABCD$ be a rectangle inscribed in circle $\Gamma$, and let $P$ be a point on the minor arc $AB$ of $\Gamma$. Suppose that $PA \cdot PB = 2$, $PC \cdot PD = 18$, and $PB \cdot PC = 9$. The area of rectangle $ABCD$ can be expressed as $\frac{a\sqrt{b}}{c}$, where $a$ and $c$ are relatively prime positive integers and $b$ is a squarefree positive integer. Compute $100a + 10b + c$.
Answer: 21055
Problem: For each nonnegative integer $n$, we define $A_n = 2^{3n} + 3^{6n+2} + 5^{6n+2}$. Find the greatest common divisor of the numbers $A_0, A_1, \ldots, A_{1999}$.
Answer: 7
Problem: Let $ABC$ be a triangle with $AB = 26$, $BC = 51$, and $CA = 73$. Let $O$ be an arbitrary point in the interior of $\triangle ABC$. Lines $\ell_1$, $\ell_2$, and $\ell_3$ pass through $O$ and are parallel to $\overline{AB}$, $\overline{BC}$, and $\overline{CA}$, respectively. The intersections of $\ell_1$, $\ell_2$, and $\ell_3$ with the sides of $\triangle ABC$ form a hexagon whose area is $A$. Compute the minimum value of $A$.
Answer: 280
Problem: The roots of the polynomial $x^4 - 4ix^3 + 3x^2 - 14ix - 44$ form the vertices of a parallelogram in the complex plane. What is the area of the parallelogram?
Answer: 12
Problem: Let $n, m$ be positive integers of different parity, and $n > m$. Find all integers $x$ such that $\frac{x^{2^n} - 1}{x^{2^m} - 1}$ is a perfect square.
Answer: 0
Problem: A cardboard box in the shape of a rectangular parallelepiped is to be enclosed in a cylindrical container with a hemispherical lid. If the total height of the container from the base to the top of the lid is $60$ centimeters and its base has a radius of $30$ centimeters, find the volume of the largest box that can be completely enclosed inside the container with the lid on.
Answer: 64000
Problem: Let $x_1 = \sqrt{10}$ and $y_1 = \sqrt{3}$. For all $n \geq 2$, let
\[
x_n = x_{n-1} \sqrt{77} + 15y_{n-1}, \\
y_n = 5x_{n-1} + y_{n-1} \sqrt{77}.
\]
Find $x_5^6 + 2x_5^4 - 9x_5^4y_5^2 - 12x_5^2y_5^2 + 27x_5^2y_5^4 + 18y_5^4 - 27y_5^6$.
Answer: 4608
Problem: Positive integer $n$ when divided by $3$ gives a remainder $a$, when divided by $5$ gives a remainder $b$, and when divided by $7$ gives a remainder $c$. Find the remainder when dividing $n$ by $105$ if $4a + 3b + 2c = 30$.
Answer: 89
Problem: Garfield and Odie are situated at $(0,0)$ and $(25,0)$, respectively. Suddenly, Garfield and Odie dash in the direction of the point $(9, 12)$ at speeds of $7$ and $10$ units per minute, respectively. During this chase, the minimum distance between Garfield and Odie can be written as $\frac{m}{\sqrt{n}}$ for relatively prime positive integers $m$ and $n$. Find $m+n$.
Answer: 159
Problem: Find the number of positive integers less than 101 that cannot be written as the difference of two squares of integers.
Answer: 25
Problem: You have four fair $6$-sided dice, each numbered $1$ to $6$ (inclusive). If all four dice are rolled, the probability that the product of the rolled numbers is prime can be written as $\frac{a}{b}$, where $a$ and $b$ are relatively prime. What is $a+b$?
Answer: 109
Problem: The diagram below shows a large triangle with area $72$. Each side of the triangle has been trisected, and line segments have been drawn between these trisection points parallel to the sides of the triangle. Find the area of the shaded region.
Answer: 48
Problem: Let $u_0, u_1, u_2, \ldots$ be integers such that $u_0 = 100$; $u_{k+2} \geq 2 + u_k$ for all $k \geq 0$; and $u_{\ell+5} \leq 5 + u_\ell$ for all $\ell \geq 0$. Find all possible values for the integer $u_{2023}$.
Answer: 2123
Problem: There are $110$ guinea pigs for each of the $110$ species, arranged in a $110 \times 110$ array. Find the maximum integer $n$ such that, no matter how the guinea pigs are arranged, we can always find a column or a row of $110$ guinea pigs containing at least $n$ different species.
Answer: 11
Problem: Compute \( \gcd \left( \frac{135^{90}-45^{90}}{90^2} , 90^2 \right) \)
Answer: 4050
Problem: S is the set of all $(a, b, c, d, e, f)$ where $a$, $b$, $c$, $d$, $e$, $f$ are integers such that $a^2 + b^2 + c^2 + d^2 + e^2 = f^2$. Find the largest $k$ which divides $abcdef$ for all members of $S$.
Answer: 24
Problem: Let $ABC$ be an isosceles triangle such that $AB = AC$ and $\angle A = 20^\circ$. Let $M$ be the foot of the altitude from $C$, and let $N$ be a point on the side $AC$ such that $CN = \frac{1}{2} BC$. Determine the measure of the angle $AMN$.
Answer: 60
Problem: 1. Let $f(n) = n^2 + 6n + 11$ be a function defined on positive integers. Find the sum of the first three prime values $f(n)$ takes on.
Answer: 753
Problem: Determine the minimum value of a prime number $p > 3$ for which there is no natural number $n > 0$ such that $2^n + 3^n \equiv 0 \pmod{p}$.
Answer: 19
Problem: The eight points $A, B, \ldots, G, H$ lie on five circles as shown. Each of these letters is represented by one of the eight numbers $1, 2, \ldots, 7, 8$, replaced such that the following conditions are met:

1. Each of the eight numbers is used exactly once.
2. The sum of the numbers on each of the five circles is the same.

How many ways are there to substitute the letters with the numbers in this way?
Answer: 8
Problem: We say that an ordered pair $(a,b)$ of positive integers with $a > b$ is square-ish if both $a+b$ and $a-b$ are perfect squares. For example, $(17,8)$ is square-ish because $17+8=25$ and $17-8=9$ are both perfect squares. How many square-ish pairs $(a,b)$ with $a+b<100$ are there?
Answer: 16
Problem: Evaluate the product \(\left(1 + \frac{2}{3}\right)\left(1 + \frac{2}{4}\right)\left(1 + \frac{2}{5}\right) \cdots \left(1 + \frac{2}{98}\right).\)
Answer: 825
Problem: Let $P(n) = (n + 1)(n + 3)(n + 5)(n + 7)(n + 9)$. What is the largest integer that is a divisor of $P(n)$ for all positive even integers $n$?
Answer: 15
Problem: Shirley has a magical machine. If she inputs a positive even integer $n$, the machine will output $\frac{n}{2}$, but if she inputs a positive odd integer $m$, the machine will output $m+3$. The machine continues operating by automatically using its output as a new input, stopping immediately before it processes a number that has already been processed. Shirley wants to create the longest possible output sequence with an initial input at most $100$. What number should she input?
Answer: 67
Problem: An infinite sequence of real numbers $a_1, a_2, \dots$ satisfies the recurrence \[ a_{n+3} = a_{n+2} - 2a_{n+1} + a_n \] for every positive integer $n$. Given that $a_1 = a_3 = 1$ and $a_{98} = a_{99}$, compute $a_1 + a_2 + \dots + a_{100}$.
Answer: 3
Problem: Triangle $ABC$ has side lengths $AB=18$, $BC=36$, and $CA=24$. The circle $\Gamma$ passes through point $C$ and is tangent to segment $AB$ at point $A$. 

Let $X$, distinct from $C$, be the second intersection of $\Gamma$ with $BC$. Moreover, let $Y$ be the point on $\Gamma$ such that segment $AY$ is an angle bisector of $\angle XAC$.

Suppose the length of segment $AY$ can be written in the form $AY=\frac{p\sqrt{r}}{q}$ where $p$, $q$, and $r$ are positive integers such that $\gcd(p, q)=1$ and $r$ is square free.

Find the value of $p+q+r$.
Answer: 69
Problem: Let $A$ and $B$ be two non-empty subsets of $X = \{1, 2, \ldots, 8 \}$ with $A \cup B = X$ and $A \cap B = \emptyset$. Let $P_A$ be the product of all elements of $A$ and let $P_B$ be the product of all elements of $B$. Find the minimum possible value of the sum $P_A + P_B$.
Answer: 402
Problem: How many positive integers $n < 500$ exist such that its prime factors are exclusively $2$, $7$, $11$, or a combination of these?
Answer: 33
Problem: Let $m$ be a positive integer less than $2015$. Suppose that the remainder when $2015$ is divided by $m$ is $n$. Compute the largest possible value of $n$.
Answer: 1007
Problem: 12 knights are sitting at a round table. Each knight is an enemy with the two knights adjacent to them, but not with any of the others. We need to choose 5 knights to save the princess such that no two knights in the group are enemies. How many ways can this be done?
Answer: 36
Problem: In the figure below, parallelograms $ABCD$ and $BFEC$ have areas $1234\, \text{cm}^2$ and $2804\, \text{cm}^2$, respectively. Points $M$ and $N$ are chosen on sides $AD$ and $FE$, respectively, so that segment $MN$ passes through $B$. Find the area of $\triangle MNC$.
Answer: 2019
Problem: The regular octagon in the figure is inscribed in a circle with a radius of $1$. Let $P$ be an arbitrary point on this circle. Calculate the value of $PA^2 + PB^2 + \ldots + PH^2$.
Answer: 16
Problem: Let $ABCD$ be a square face of a cube with edge length $2$. A plane $P$ that contains point $A$ and the midpoint of line segment $\overline{BC}$ splits the cube into two pieces of equal volume. What is the square of the area of the intersection of plane $P$ and the cube?
Answer: 24
Problem: Let $\frac{a}{b}$ be a fraction such that $a$ and $b$ are positive integers and the first three digits of its decimal expansion are 527. What is the smallest possible value of $a+b?$
Answer: 55
Problem: If we divide the number $19250$ by a certain number, the remainder is $11$. Similarly, if we divide the number $20302$ by the same number, the remainder is $3$. What is this number?
Answer: 53
Problem: Let $AB$ be a chord of a circle with center $O$. Let $C$ be a point on the circle such that $\angle ABC = 30^\circ$ and $O$ lies inside the triangle $ABC$. Let $D$ be a point on $AB$ such that $\angle DCO = \angle OCB = 20^\circ$. Find the measure of $\angle CDO$ in degrees.
Answer: 80
Problem: Lucy has $8$ children, each of whom has a distinct favorite integer from $1$ to $10$, inclusive. The smallest number that is a perfect multiple of all of these favorite numbers is $1260$, and the average of these favorite numbers is at most $5$. Find the sum of the four largest numbers.
Answer: 27
Problem: Suppose there are 36 objects placed at equal distances along a circle. Determine the number of ways to choose 3 objects such that no two chosen objects are adjacent or diametrically opposite.
Answer: 5412
Problem: A pen costs \(11\) € and a notebook costs \(13\) €. Find the number of ways in which a person can spend exactly \(1000\) € to buy pens and notebooks.
Answer: 7
Problem: Define $H_n = 1+\frac{1}{2}+\cdots+\frac{1}{n}$. Let the sum of all $H_n$ that are terminating in base 10 be $S$. If $S = \frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers, find $100m+n$.
Answer: 9920
Problem: Let $S = \{1, 22, 333, \dots , 999999999\}$. For how many pairs of integers $(a, b)$ where $a, b \in S$ and $a < b$ is it the case that $a$ divides $b$?
Answer: 14
Problem: If we define $\otimes(a,b,c)$ by
\begin{align*}
\otimes(a,b,c) = \frac{\max(a,b,c)- \min(a,b,c)}{a+b+c-\min(a,b,c)-\max(a,b,c)},
\end{align*}
compute $\otimes(\otimes(7,1,3),\otimes(-3,-4,2),1)$.
Answer: 4
Problem: In a city, there are $2019$ metro stations. Some pairs of stations are connected by tunnels, and from any station, you can reach any other station through these tunnels. The mayor ordered the organization of several metro lines: each line should include several different stations connected in series by tunnels (several lines can pass through the same tunnel), and each station must be on at least one line. To save money, no more than $k$ lines should be made. It turned out that the mayor's order is not feasible. What is the largest value of $k$ for which this could happen?
Answer: 1008
Problem: What is the $22^{\text{nd}}$ positive integer $n$ such that $22^n$ ends in a $2$ (when written in base $10$)?
Answer: 85
Problem: Determine the last three digits of $2003^{2002^{2001}}$.
Answer: 241
Problem: For a positive integer $k$, let $f_1(k)$ be the square of the sum of the digits of $k$. Define $f_{n+1} = f_1 \circ f_n$. Evaluate $f_{2007}(2^{2006})$.
Answer: 169
Problem: An elephant writes a sequence of numbers on a board starting with $1$. Each minute, it doubles the sum of all the numbers on the board so far, and without erasing anything, writes the result on the board. It stops after writing a number greater than one billion. How many distinct prime factors does the largest number on the board have?
Answer: 2
Problem: Add one pair of brackets to the expression:

\[1 + 2 \times 3 + 4 \times 5 + 6\]

so that the resulting expression has a valid mathematical value. For example, \(1 + 2 \times (3 + 4 \times 5) + 6 = 53\). What is the largest possible value that one can make?
Answer: 77
Problem: Determine the smallest positive integer containing only $0$ and $1$ as digits that is divisible by each integer from $1$ through $9$.
Answer: 1111111110