ChenDRAG/VeRL_math_validation huggingface dataset:
Problem: On a flat plane in Camelot, King Arthur builds a labyrinth $\mathfrak{L}$ consisting of $n$ walls, each of which is an infinite straight line. No two walls are parallel, and no three walls have a common point. Merlin then paints one side of each wall entirely red and the other side entirely blue.

At the intersection of two walls there are four corners: two diagonally opposite corners where a red side and a blue side meet, one corner where two red sides meet, and one corner where two blue sides meet. At each such intersection, there is a two-way door connecting the two diagonally opposite corners at which sides of different colours meet.

After Merlin paints the walls, Morgana then places some knights in the labyrinth. The knights can walk through doors, but cannot walk through walls.

Let $k(\mathfrak{L})$ be the largest number $k$ such that, no matter how Merlin paints the labyrinth $\mathfrak{L}$, Morgana can always place at least $k$ knights such that no two of them can ever meet. For each $n$, what are all possible values for $k(\mathfrak{L})$, where $\mathfrak{L}$ is a labyrinth with $n$ walls?
Answer: k=n+1
Problem: There are 60 empty boxes $B_{1}, \ldots, B_{60}$ in a row on a table and an unlimited supply of pebbles. Given a positive integer $n$, Alice and Bob play the following game.

In the first round, Alice takes $n$ pebbles and distributes them into the 60 boxes as she wishes. Each subsequent round consists of two steps:

(a) Bob chooses an integer $k$ with $1 \leqslant k \leqslant 59$ and splits the boxes into the two groups $B_{1}, \ldots, B_{k}$ and $B_{k+1}, \ldots, B_{60}$.

(b) Alice picks one of these two groups, adds one pebble to each box in that group, and removes one pebble from each box in the other group.

Bob wins if, at the end of any round, some box contains no pebbles. Find the smallest $n$ such that Alice can prevent Bob from winning.
Answer: 960
Problem: For any two different real numbers $x$ and $y$, we define $D(x, y)$ to be the unique integer $d$ satisfying $2^{d} \leqslant|x-y|<2^{d+1}$. Given a set of reals $\mathcal{F}$, and an element $x \in \mathcal{F}$, we say that the scales of $x$ in $\mathcal{F}$ are the values of $D(x, y)$ for $y \in \mathcal{F}$ with $x \neq y$.

Let $k$ be a given positive integer. Suppose that each member $x$ of $\mathcal{F}$ has at most $k$ different scales in $\mathcal{F}$ (note that these scales may depend on $x$ ). What is the maximum possible size of $\mathcal{F}$ ?
Answer: 2^{k}
Problem: Find all pairs $(m, n)$ of positive integers satisfying the equation

$$
\left(2^{n}-1\right)\left(2^{n}-2\right)\left(2^{n}-4\right) \cdots\left(2^{n}-2^{n-1}\right)=m !
\tag{1}
$$
Answer: (1,1), (3,2)
Problem: Find all triples $(a, b, c)$ of positive integers such that $a^{3}+b^{3}+c^{3}=(a b c)^{2}$.
Answer: (1,2,3),(1,3,2),(2,1,3),(2,3,1),(3,1,2),(3,2,1)
Problem: Determine all functions $f: \mathbb{Z} \rightarrow \mathbb{Z}$ with the property that

$$
f(x-f(y))=f(f(x))-f(y)-1
\tag{1}
$$

holds for all $x, y \in \mathbb{Z}$.
Answer: f(x)=-1$,$f(x)=x+1
Problem: Let $n$ be a fixed positive integer. Find the maximum possible value of

$$
\sum_{1 \leqslant r<s \leqslant 2 n}(s-r-n) x_{r} x_{s}
$$

where $-1 \leqslant x_{i} \leqslant 1$ for all $i=1,2, \ldots, 2 n$.
Answer: n(n-1)
Problem: Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ satisfying the equation

$$
f(x+f(x+y))+f(x y)=x+f(x+y)+y f(x)\tag{1}
$$

for all real numbers $x$ and $y$.
Answer: f(x)=x$,$f(x)=2-x
Problem: For a finite set $A$ of positive integers, we call a partition of $A$ into two disjoint nonempty subsets $A_{1}$ and $A_{2}$ good if the least common multiple of the elements in $A_{1}$ is equal to the greatest common divisor of the elements in $A_{2}$. Determine the minimum value of $n$ such that there exists a set of $n$ positive integers with exactly 2015 good partitions.
Answer: 3024
Problem: Let $A B C$ be an acute triangle, and let $M$ be the midpoint of $A C$. A circle $\omega$ passing through $B$ and $M$ meets the sides $A B$ and $B C$ again at $P$ and $Q$, respectively. Let $T$ be the point such that the quadrilateral $B P T Q$ is a parallelogram. Suppose that $T$ lies on the circumcircle of the triangle $A B C$. Determine all possible values of $B T / B M$.
Answer: \sqrt{2}
Problem: Determine all triples $(a, b, c)$ of positive integers for which $a b-c, b c-a$, and $c a-b$ are powers of 2 .

Explanation: A power of 2 is an integer of the form $2^{n}$, where $n$ denotes some nonnegative integer.
Answer: (2,2,2),(2,2,3),(2,3,2),(3,2,2),(2,6,11),(2,11,6),(6,2,11),(6,11,2),(11,2,6),(11,6,2),(3,5,7),(3,7,5),(5,3,7),(5,7,3),(7,3,5),(7,5,3)
Problem: Let $\mathbb{Z}_{>0}$ denote the set of positive integers. For any positive integer $k$, a function $f: \mathbb{Z}_{>0} \rightarrow \mathbb{Z}_{>0}$ is called $k$-good if $\operatorname{gcd}(f(m)+n, f(n)+m) \leqslant k$ for all $m \neq n$. Find all $k$ such that there exists a $k$-good function.
Answer: k \geqslant 2
Problem: For every positive integer $n$ with prime factorization $n=\prod_{i=1}^{k} p_{i}^{\alpha_{i}}$, define

$$
\mho(n)=\sum_{i: p_{i}>10^{100}} \alpha_{i}\tag{1}
$$

That is, $\mho(n)$ is the number of prime factors of $n$ greater than $10^{100}$, counted with multiplicity.

Find all strictly increasing functions $f: \mathbb{Z} \rightarrow \mathbb{Z}$ such that

$$
\mho(f(a)-f(b)) \leqslant \mho(a-b) \quad \text { for all integers } a \text { and } b \text { with } a>b \text {. }
$$
Answer: f(x)=a x+b$, where $b$ is an arbitrary integer, and $a$ is an arbitrary positive integer with $\mho(a)=0
Problem: For a sequence $x_{1}, x_{2}, \ldots, x_{n}$ of real numbers, we define its price as

$$
\max _{1 \leqslant i \leqslant n}\left|x_{1}+\cdots+x_{i}\right|
$$

Given $n$ real numbers, Dave and George want to arrange them into a sequence with a low price. Diligent Dave checks all possible ways and finds the minimum possible price $D$. Greedy George, on the other hand, chooses $x_{1}$ such that $\left|x_{1}\right|$ is as small as possible; among the remaining numbers, he chooses $x_{2}$ such that $\left|x_{1}+x_{2}\right|$ is as small as possible, and so on. Thus, in the $i^{\text {th }}$ step he chooses $x_{i}$ among the remaining numbers so as to minimise the value of $\left|x_{1}+x_{2}+\cdots+x_{i}\right|$. In each step, if several numbers provide the same value, George chooses one at random. Finally he gets a sequence with price $G$.

Find the least possible constant $c$ such that for every positive integer $n$, for every collection of $n$ real numbers, and for every possible sequence that George might obtain, the resulting values satisfy the inequality $G \leqslant c D$.
Answer: 2
Problem: Determine all functions $f: \mathbb{Z} \rightarrow \mathbb{Z}$ satisfying

$$
f(f(m)+n)+f(m)=f(n)+f(3 m)+2014
\tag{1}
$$

for all integers $m$ and $n$.
Answer: f(n) = 2 n+1007
Problem: Consider all polynomials $P(x)$ with real coefficients that have the following property: for any two real numbers $x$ and $y$ one has

$$
\left|y^{2}-P(x)\right| \leqslant 2|x| \text { if and only if }\left|x^{2}-P(y)\right| \leqslant 2|y|
\tag{1}
$$

Determine all possible values of $P(0)$.
Answer: (-\infty, 0) \cup\{1\}$.
Problem: Let $n \geqslant 2$ be an integer. Consider an $n \times n$ chessboard divided into $n^{2}$ unit squares. We call a configuration of $n$ rooks on this board happy if every row and every column contains exactly one rook. Find the greatest positive integer $k$ such that for every happy configuration of rooks, we can find a $k \times k$ square without a rook on any of its $k^{2}$ unit squares.
Answer: \lfloor\sqrt{n-1}\rfloor
Problem: We are given an infinite deck of cards, each with a real number on it. For every real number $x$, there is exactly one card in the deck that has $x$ written on it. Now two players draw disjoint sets $A$ and $B$ of 100 cards each from this deck. We would like to define a rule that declares one of them a winner. This rule should satisfy the following conditions:

1. The winner only depends on the relative order of the 200 cards: if the cards are laid down in increasing order face down and we are told which card belongs to which player, but not what numbers are written on them, we can still decide the winner.
2. If we write the elements of both sets in increasing order as $A=\left\{a_{1}, a_{2}, \ldots, a_{100}\right\}$ and $B=\left\{b_{1}, b_{2}, \ldots, b_{100}\right\}$, and $a_{i}>b_{i}$ for all $i$, then $A$ beats $B$.
3. If three players draw three disjoint sets $A, B, C$ from the deck, $A$ beats $B$ and $B$ beats $C$, then $A$ also beats $C$.

How many ways are there to define such a rule? Here, we consider two rules as different if there exist two sets $A$ and $B$ such that $A$ beats $B$ according to one rule, but $B$ beats $A$ according to the other.
Answer: 100
Problem: Let $n \geqslant 2$ be an integer, and let $A_{n}$ be the set

$$
A_{n}=\left\{2^{n}-2^{k} \mid k \in \mathbb{Z}, 0 \leqslant k<n\right\} .
$$

Determine the largest positive integer that cannot be written as the sum of one or more (not necessarily distinct) elements of $A_{n}$.
Answer: (n-2) 2^{n}+1
Problem: Let $k \geqslant 2$ be an integer. Find the smallest integer $n \geqslant k+1$ with the property that there exists a set of $n$ distinct real numbers such that each of its elements can be written as a sum of $k$ other distinct elements of the set.
Answer: n=k+4
Problem: Let $\mathbb{R}_{>0}$ be the set of positive real numbers. Find all functions $f: \mathbb{R}_{>0} \rightarrow \mathbb{R}_{>0}$ such that, for every $x \in \mathbb{R}_{>0}$, there exists a unique $y \in \mathbb{R}_{>0}$ satisfying

$$
x f(y)+y f(x) \leqslant 2 .
$$
Answer: f(x)=\frac{1}{x}
Problem: Find all positive integers $n \geqslant 2$ for which there exist $n$ real numbers $a_{1}<\cdots<a_{n}$ and a real number $r>0$ such that the $\frac{1}{2} n(n-1)$ differences $a_{j}-a_{i}$ for $1 \leqslant i<j \leqslant n$ are equal, in some order, to the numbers $r^{1}, r^{2}, \ldots, r^{\frac{1}{2} n(n-1)}$.
Answer: 2,3,4
Problem: $A \pm 1 \text{-}sequence$ is a sequence of 2022 numbers $a_{1}, \ldots, a_{2022}$, each equal to either +1 or -1 . Determine the largest $C$ so that, for any $\pm 1 -sequence$, there exists an integer $k$ and indices $1 \leqslant t_{1}<\ldots<t_{k} \leqslant 2022$ so that $t_{i+1}-t_{i} \leqslant 2$ for all $i$, and

$$
\left|\sum_{i=1}^{k} a_{t_{i}}\right| \geqslant C
$$
Answer: 506
Problem: In each square of a garden shaped like a $2022 \times 2022$ board, there is initially a tree of height 0 . A gardener and a lumberjack alternate turns playing the following game, with the gardener taking the first turn:

- The gardener chooses a square in the garden. Each tree on that square and all the surrounding squares (of which there are at most eight) then becomes one unit taller.
- The lumberjack then chooses four different squares on the board. Each tree of positive height on those squares then becomes one unit shorter.

We say that a tree is majestic if its height is at least $10^{6}$. Determine the largest number $K$ such that the gardener can ensure there are eventually $K$ majestic trees on the board, no matter how the lumberjack plays.
Answer: 2271380
Problem: Lucy starts by writing $s$ integer-valued 2022-tuples on a blackboard. After doing that, she can take any two (not necessarily distinct) tuples $\mathbf{v}=\left(v_{1}, \ldots, v_{2022}\right)$ and $\mathbf{w}=\left(w_{1}, \ldots, w_{2022}\right)$ that she has already written, and apply one of the following operations to obtain a new tuple:

$$
\begin{aligned}
& \mathbf{v}+\mathbf{w}=\left(v_{1}+w_{1}, \ldots, v_{2022}+w_{2022}\right) \\
& \mathbf{v} \vee \mathbf{w}=\left(\max \left(v_{1}, w_{1}\right), \ldots, \max \left(v_{2022}, w_{2022}\right)\right)
\end{aligned}
$$

and then write this tuple on the blackboard.

It turns out that, in this way, Lucy can write any integer-valued 2022-tuple on the blackboard after finitely many steps. What is the smallest possible number $s$ of tuples that she initially wrote?
Answer: 3
Problem: Alice fills the fields of an $n \times n$ board with numbers from 1 to $n^{2}$, each number being used exactly once. She then counts the total number of good paths on the board. A good path is a sequence of fields of arbitrary length (including 1) such that:

(i) The first field in the sequence is one that is only adjacent to fields with larger numbers,

(ii) Each subsequent field in the sequence is adjacent to the previous field,

(iii) The numbers written on the fields in the sequence are in increasing order.

Two fields are considered adjacent if they share a common side. Find the smallest possible number of good paths Alice can obtain, as a function of $n$.
Answer: 2 n^{2}-2 n+1
Problem: Let $\mathbb{Z}_{\geqslant 0}$ be the set of non-negative integers, and let $f: \mathbb{Z}_{\geqslant 0} \times \mathbb{Z}_{\geqslant 0} \rightarrow \mathbb{Z}_{\geqslant 0}$ be a bijection such that whenever $f\left(x_{1}, y_{1}\right)>f\left(x_{2}, y_{2}\right)$, we have $f\left(x_{1}+1, y_{1}\right)>f\left(x_{2}+1, y_{2}\right)$ and $f\left(x_{1}, y_{1}+1\right)>f\left(x_{2}, y_{2}+1\right)$.

Let $N$ be the number of pairs of integers $(x, y)$, with $0 \leqslant x, y<100$, such that $f(x, y)$ is odd. Find the smallest and largest possible value of $N$.
Answer: 2500,7500
Problem: A number is called Norwegian if it has three distinct positive divisors whose sum is equal to 2022. Determine the smallest Norwegian number.

(Note: The total number of positive divisors of a Norwegian number is allowed to be larger than 3.)
Answer: 1344
Problem: Find all positive integers $n>2$ such that

$$
n ! \mid \prod_{\substack{p<q \leqslant n \\ p, q \text { primes }}}(p+q)
$$
Answer: 7
Problem: Find all triples of positive integers $(a, b, p)$ with $p$ prime and

$$
a^{p}=b !+p
$$
Answer: (2,2,2),(3,4,3)
Problem: Let $\mathbb{Q}_{>0}$ denote the set of all positive rational numbers. Determine all functions $f: \mathbb{Q}_{>0} \rightarrow \mathbb{Q}_{>0}$ satisfying

$$
f\left(x^{2} f(y)^{2}\right)=f(x)^{2} f(y)
\tag{*}
$$

for all $x, y \in \mathbb{Q}_{>0}$.
Answer: f(x)=1
Problem: Let $a_{0}, a_{1}, a_{2}, \ldots$ be a sequence of real numbers such that $a_{0}=0, a_{1}=1$, and for every $n \geqslant 2$ there exists $1 \leqslant k \leqslant n$ satisfying

$$
a_{n}=\frac{a_{n-1}+\cdots+a_{n-k}}{k}
$$

Find the maximal possible value of $a_{2018}-a_{2017}$.
Answer: \frac{2016}{2017^{2}}
Problem: Find the maximal value of

$$
S=\sqrt[3]{\frac{a}{b+7}}+\sqrt[3]{\frac{b}{c+7}}+\sqrt[3]{\frac{c}{d+7}}+\sqrt[3]{\frac{d}{a+7}}
$$

where $a, b, c, d$ are nonnegative real numbers which satisfy $a+b+c+d=100$.
Answer: \frac{8}{\sqrt[3]{7}}
Problem: Queenie and Horst play a game on a $20 \times 20$ chessboard. In the beginning the board is empty. In every turn, Horst places a black knight on an empty square in such a way that his new knight does not attack any previous knights. Then Queenie places a white queen on an empty square. The game gets finished when somebody cannot move.

Find the maximal positive $K$ such that, regardless of the strategy of Queenie, Horst can put at least $K$ knights on the board.
Answer: 100
Problem: Let $k$ be a positive integer. The organising committee of a tennis tournament is to schedule the matches for $2 k$ players so that every two players play once, each day exactly one match is played, and each player arrives to the tournament site the day of his first match, and departs the day of his last match. For every day a player is present on the tournament, the committee has to pay 1 coin to the hotel. The organisers want to design the schedule so as to minimise the total cost of all players' stays. Determine this minimum cost.
Answer: k\left(4 k^{2}+k-1\right) / 2
Problem: A circle $\omega$ of radius 1 is given. A collection $T$ of triangles is called good, if the following conditions hold:

(i) each triangle from $T$ is inscribed in $\omega$;

(ii) no two triangles from $T$ have a common interior point.

Determine all positive real numbers $t$ such that, for each positive integer $n$, there exists a good collection of $n$ triangles, each of perimeter greater than $t$.
Answer: t(0,4]
Problem: Let $n$ be a positive integer. Find the smallest integer $k$ with the following property: Given any real numbers $a_{1}, \ldots, a_{d}$ such that $a_{1}+a_{2}+\cdots+a_{d}=n$ and $0 \leqslant a_{i} \leqslant 1$ for $i=1,2, \ldots, d$, it is possible to partition these numbers into $k$ groups (some of which may be empty) such that the sum of the numbers in each group is at most 1 .
Answer: k=2 n-1
Problem: In the plane, 2013 red points and 2014 blue points are marked so that no three of the marked points are collinear. One needs to draw $k$ lines not passing through the marked points and dividing the plane into several regions. The goal is to do it in such a way that no region contains points of both colors.

Find the minimal value of $k$ such that the goal is attainable for every possible configuration of 4027 points.
Answer: 2013
Problem: Let $\mathbb{Z}_{>0}$ be the set of positive integers. Find all functions $f: \mathbb{Z}_{>0} \rightarrow \mathbb{Z}_{>0}$ such that

$$
m^{2}+f(n) \mid m f(m)+n
$$

for all positive integers $m$ and $n$.
Answer: f(n)=n
Problem: Find the largest possible integer $k$, such that the following statement is true:

Let 2009 arbitrary non-degenerated triangles be given. In every triangle the three sides are colored, such that one is blue, one is red and one is white. Now, for every color separately, let us sort the lengths of the sides. We obtain

$$
\begin{aligned}
b_{1} \leq b_{2} \leq \ldots \leq b_{2009} & \text { the lengths of the blue sides } \\
r_{1} \leq r_{2} \leq \ldots \leq r_{2009} & \text { the lengths of the red sides, } \\
\text { and } \quad & w_{1} \leq w_{2} \leq \ldots \leq w_{2009} \quad \text { the lengths of the white sides. }
\end{aligned}
$$

Then there exist $k$ indices $j$ such that we can form a non-degenerated triangle with side lengths $b_{j}, r_{j}, w_{j}$.
Answer: 1
Problem: Determine all functions $f$ from the set of positive integers into the set of positive integers such that for all $x$ and $y$ there exists a non degenerated triangle with sides of lengths

$$
x, \quad f(y) \text { and } f(y+f(x)-1) .
$$
Answer: f(z)=z
Problem: For any integer $n \geq 2$, let $N(n)$ be the maximal number of triples $\left(a_{i}, b_{i}, c_{i}\right), i=1, \ldots, N(n)$, consisting of nonnegative integers $a_{i}, b_{i}$ and $c_{i}$ such that the following two conditions are satisfied:

(1) $a_{i}+b_{i}+c_{i}=n$ for all $i=1, \ldots, N(n)$,

(2) If $i \neq j$, then $a_{i} \neq a_{j}, b_{i} \neq b_{j}$ and $c_{i} \neq c_{j}$.

Determine $N(n)$ for all $n \geq 2$.
Answer: N(n)=\left\lfloor\frac{2 n}{3}\right\rfloor+1
Problem: On a $999 \times 999$ board a limp rook can move in the following way: From any square it can move to any of its adjacent squares, i.e. a square having a common side with it, and every move must be a turn, i.e. the directions of any two consecutive moves must be perpendicular. A nonintersecting route of the limp rook consists of a sequence of pairwise different squares that the limp rook can visit in that order by an admissible sequence of moves. Such a non-intersecting route is called cyclic, if the limp rook can, after reaching the last square of the route, move directly to the first square of the route and start over.

How many squares does the longest possible cyclic, non-intersecting route of a limp rook visit?
Answer: 996000
Problem: Let $A B C$ be a triangle with $A B=A C$. The angle bisectors of $A$ and $B$ meet the sides $B C$ and $A C$ in $D$ and $E$, respectively. Let $K$ be the incenter of triangle $A D C$. Suppose that $\angle B E K=45^{\circ}$. Find all possible values of $\angle B A C$.
Answer: 90^{\circ}$,$60^{\circ}
Problem: Find all positive integers $n$ such that there exists a sequence of positive integers $a_{1}, a_{2}, \ldots, a_{n}$ satisfying

$$
a_{k+1}=\frac{a_{k}^{2}+1}{a_{k-1}+1}-1
$$

for every $k$ with $2 \leq k \leq n-1$.
Answer: 1,2,3,4
Problem: In the plane we consider rectangles whose sides are parallel to the coordinate axes and have positive length. Such a rectangle will be called a box. Two boxes intersect if they have a common point in their interior or on their boundary.

Find the largest $n$ for which there exist $n$ boxes $B_{1}, \ldots, B_{n}$ such that $B_{i}$ and $B_{j}$ intersect if and only if $i \not \equiv j \pm 1(\bmod n)$.
Answer: 6
Problem: In the coordinate plane consider the set $S$ of all points with integer coordinates. For a positive integer $k$, two distinct points $A, B \in S$ will be called $k$-friends if there is a point $C \in S$ such that the area of the triangle $A B C$ is equal to $k$. A set $T \subset S$ will be called a $k$-clique if every two points in $T$ are $k$-friends. Find the least positive integer $k$ for which there exists a $k$-clique with more than 200 elements.
Answer: 180180
Problem: Let $n$ and $k$ be fixed positive integers of the same parity, $k \geq n$. We are given $2 n$ lamps numbered 1 through $2 n$; each of them can be on or off. At the beginning all lamps are off. We consider sequences of $k$ steps. At each step one of the lamps is switched (from off to on or from on to off).

Let $N$ be the number of $k$-step sequences ending in the state: lamps $1, \ldots, n$ on, lamps $n+1, \ldots, 2 n$ off.

Let $M$ be the number of $k$-step sequences leading to the same state and not touching lamps $n+1, \ldots, 2 n$ at all.

Find the ratio $N / M$.
Answer: 2^{k-n}
Problem: Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ that satisfy the conditions

$$
f(1+x y)-f(x+y)=f(x) f(y) \text { for all } x, y \in \mathbb{R}
$$

and $f(-1) \neq 0$.
Answer: f(x)=x-1
Problem: Let $n \geq 1$ be an integer. What is the maximum number of disjoint pairs of elements of the set $\{1,2, \ldots, n\}$ such that the sums of the different pairs are different integers not exceeding $n$ ?
Answer: \lfloor\frac{2 n-1}{5}\rfloor
Problem: In a $999 \times 999$ square table some cells are white and the remaining ones are red. Let $T$ be the number of triples $\left(C_{1}, C_{2}, C_{3}\right)$ of cells, the first two in the same row and the last two in the same column, with $C_{1}$ and $C_{3}$ white and $C_{2}$ red. Find the maximum value $T$ can attain.
Answer: \frac{4 \cdot 999^{4}}{27}
Problem: Players $A$ and $B$ play a game with $N \geq 2012$ coins and 2012 boxes arranged around a circle. Initially $A$ distributes the coins among the boxes so that there is at least 1 coin in each box. Then the two of them make moves in the order $B, A, B, A, \ldots$ by the following rules:

- On every move of his $B$ passes 1 coin from every box to an adjacent box.
- On every move of hers $A$ chooses several coins that were not involved in $B$ 's previous move and are in different boxes. She passes every chosen coin to an adjacent box.

Player $A$ 's goal is to ensure at least 1 coin in each box after every move of hers, regardless of how $B$ plays and how many moves are made. Find the least $N$ that enables her to succeed.
Answer: 4022
Problem: Find all triples $(x, y, z)$ of positive integers such that $x \leq y \leq z$ and

$$
x^{3}\left(y^{3}+z^{3}\right)=2012(x y z+2) \text {. }
$$
Answer: (2,251,252)
Problem: Find all functions $f: \mathbb{Q} \rightarrow \mathbb{Q}$ such that the equation

holds for all rational numbers $x$ and $y$.

$$
f(x f(x)+y)=f(y)+x^{2}
$$

Here, $\mathbb{Q}$ denotes the set of rational numbers.
Answer: f(x)=x,f(x)=-x
Problem: A plane has a special point $O$ called the origin. Let $P$ be a set of 2021 points in the plane, such that

(i) no three points in $P$ lie on a line and

(ii) no two points in $P$ lie on a line through the origin.

A triangle with vertices in $P$ is $f a t$, if $O$ is strictly inside the triangle. Find the maximum number of fat triangles.
Answer: 2021 \cdot 505 \cdot 337