Let $N$ be a positive integer, and consider an $N \times N$ grid. A [i]right-down path[/i] is a sequence of grid cells such that each cell is either one cell to the right of or one cell below the previous cell in the sequence. A [i]right-up path[/i] is a sequence of grid cells such that each cell is either one cell to the right of or one cell above the previous cell in the sequence. Prove that the cells of the $N \times N$ grid cannot be partitioned into less than $N$ right-down or right-up paths. For example, the following partition of the $5 \times 5$ grid uses $5$ paths. [asy] size(4cm); draw((5,-1)--(0,-1)--(0,-2)--(5,-2)--(5,-3)--(0,-3)--(0,-4)--(5,-4),gray+linewidth(0.5)+miterjoin); draw((1,-5)--(1,0)--(2,0)--(2,-5)--(3,-5)--(3,0)--(4,0)--(4,-5),gray+linewidth(0.5)+miterjoin); draw((0,0)--(5,0)--(5,-5)--(0,-5)--cycle,black+linewidth(2.5)+miterjoin); draw((0,-1)--(3,-1)--(3,-2)--(1,-2)--(1,-4)--(4,-4)--(4,-3)--(2,-3)--(2,-2),black+linewidth(2.5)+miterjoin); draw((3,0)--(3,-1),black+linewidth(2.5)+miterjoin); draw((1,-4)--(1,-5),black+linewidth(2.5)+miterjoin); draw((4,-3)--(4,-1)--(5,-1),black+linewidth(2.5)+miterjoin); [/asy] [i]Proposed by Zixiang Zhou, Canada[/i]

We say that the prime numbers $p_1,\dots,p_n$ construct the graph $G$ if we can assign to each vertex of $G$ a natural number whose prime divisors are among $p_1,\dots,p_n$ and there is an edge between two vertices in $G$ if and only if the numbers assigned to the two vertices have a common divisor greater than $1$. What is the minimal $n$ such that there exist prime numbers $p_1,\dots,p_n$ which construct any graph $G$ with $N$ vertices?

Let $n$ be a positive integer. Define a chameleon to be any sequence of $3n$ letters, with exactly $n$ occurrences of each of the letters $a, b,$ and $c$. Define a swap to be the transposition of two adjacent letters in a chameleon. Prove that for any chameleon $X$ , there exists a chameleon $Y$ such that $X$ cannot be changed to $Y$ using fewer than $3n^2/2$ swaps.

There are $n$ points on a circle ($n>1$). Define an "interval" as an arc of a circle such that it's start and finish are from those points. Consider a family of intervals $F$ such that for every element of $F$ like $A$ there is almost one other element of $F$ like $B$ such that $A \subseteq B$ (in this case we call $A$ is sub-interval of $B$). We call an interval maximal if it is not a sub-interval of any other interval. If $m$ is the number of maximal elements of $F$ and $a$ is number of non-maximal elements of $F,$ prove that $n\geq m+\frac a2.$

Let $\mathbb{Z}_{>0}$ denote the set of positive integers. Consider a function $f: \mathbb{Z}_{>0} \to \mathbb{Z}_{>0}$. For any $m, n \in \mathbb{Z}_{>0}$ we write $f^n(m) = \underbrace{f(f(\ldots f}_{n}(m)\ldots))$. Suppose that $f$ has the following two properties: (i) if $m, n \in \mathbb{Z}_{>0}$, then $\frac{f^n(m) - m}{n} \in \mathbb{Z}_{>0}$; (ii) The set $\mathbb{Z}_{>0} \setminus \{f(n) \mid n\in \mathbb{Z}_{>0}\}$ is finite. Prove that the sequence $f(1) - 1, f(2) - 2, f(3) - 3, \ldots$ is periodic. [i]Proposed by Ang Jie Jun, Singapore[/i]

We say that a polygon is rectangular when all of its angles are $90^\circ$ or $270^\circ$. Is it true that each rectangular polygon, which sides are with length equal to odd numbers only, [u]can't[/u] be covered with 2x1 domino tiles?

For each prime $p$, construct a graph $G_p$ on $\{1,2,\ldots p\}$, where $m\neq n$ are adjacent if and only if $p$ divides $(m^{2} + 1-n)(n^{2} + 1-m)$. Prove that $G_p$ is disconnected for infinitely many $p$

For each prime $p$, construct a graph $G_p$ on $\{1,2,\ldots p\}$, where $m\neq n$ are adjacent if and only if $p$ divides $(m^{2} + 1-n)(n^{2} + 1-m)$. Prove that $G_p$ is disconnected for infinitely many $p$

$a,b,c$ are positive numbers such that $ a^2 + b^2 + c^2 = 2abc + 1 $. Find the maximum value of \[ (a-2bc)(b-2ca)(c-2ab) \]

A [i]site[/i] is any point $(x, y)$ in the plane such that $x$ and $y$ are both positive integers less than or equal to 20. Initially, each of the 400 sites is unoccupied. Amy and Ben take turns placing stones with Amy going first. On her turn, Amy places a new red stone on an unoccupied site such that the distance between any two sites occupied by red stones is not equal to $\sqrt{5}$. On his turn, Ben places a new blue stone on any unoccupied site. (A site occupied by a blue stone is allowed to be at any distance from any other occupied site.) They stop as soon as a player cannot place a stone. Find the greatest $K$ such that Amy can ensure that she places at least $K$ red stones, no matter how Ben places his blue stones. [i]Proposed by Gurgen Asatryan, Armenia[/i]

A group consist of n tourists. Among every 3 of them there are 2 which are not familiar. For every partition of the tourists in 2 buses you can find 2 tourists that are in the same bus and are familiar with each other. Prove that is a tourist familiar to at most $\displaystyle \frac 2{5}n$ tourists.

There are $300$ participants to a mathematics competition. After the competition some of the contestants play some games of chess. Each two contestants play at most one game against each other. There are no three contestants, such that each of them plays against each other. Determine the maximum value of $n$ for which it is possible to satisfy the following conditions at the same time: each contestant plays at most $n$ games of chess, and for each $m$ with $1 \le m \le n$, there is a contestant playing exactly $m$ games of chess.

A function $\psi \colon {\mathbb Z} \to {\mathbb Z}$ is said to be [i]zero-requiem[/i] if for any positive integer $n$ and any integers $a_1$, $\ldots$, $a_n$ (not necessarily distinct), the sums $a_1 + a_2 + \dots + a_n$ and $\psi(a_1) + \psi(a_2) + \dots + \psi(a_n)$ are not both zero. Let $f$ and $g$ be two zero-requiem functions for which $f \circ g$ and $g \circ f$ are both the identity function (that is, $f$ and $g$ are mutually inverse bijections). Given that $f+g$ is [i]not[/i] a zero-requiem function, prove that $f \circ f$ and $g \circ g$ are both zero-requiem. [i]Sutanay Bhattacharya[/i]

There are $n>1$ cities in the country, some pairs of cities linked two-way through straight flight. For every pair of cities there is exactly one aviaroute (can have interchanges). Major of every city X counted amount of such numberings of all cities from $1$ to $n$ , such that on every aviaroute with the beginning in X, numbers of cities are in ascending order. Every major, except one, noticed that results of counting are multiple of $2016$. Prove, that result of last major is multiple of $2016$ too.