Call admissible a set $A$ of integers that has the following property: If $x,y \in A$ (possibly $x=y$) then $x^2+kxy+y^2 \in A$ for every integer $k$. Determine all pairs $m,n$ of nonzero integers such that the only admissible set containing both $m$ and $n$ is the set of all integers. [i]Proposed by Warut Suksompong, Thailand[/i]

Jim and Jane divide a triangular cake between themselves. Jim chooses any point in the cake and Jane makes a straight cut through this point and chooses the piece. Find the size of the piece that each of them can guarantee for himself/herself (both of them want to get as much as possible). [i](4 points)[/i]

A triangular pyramid $ABCD$ is given. A sphere $\omega_A$ is tangent to the face $BCD$ and to the planes of other faces in points don't lying on faces. Similarly, sphere $\omega_B$ is tangent to the face $ACD$ and to the planes of other faces in points don't lying on faces. Let $K$ be the point where $\omega_A$ is tangent to $ACD$, and let $L$ be the point where $\omega_B$ is tangent to $BCD$. The points $X$ and $Y$ are chosen on the prolongations of $AK$ and $BL$ over $K$ and $L$ such that $\angle CKD = \angle CXD + \angle CBD$ and $\angle CLD = \angle CYD +\angle CAD$. Prove that the distances from the points $X$, $Y$ to the midpoint of $CD$ are the same. [hide=thanks ]Thanks to the user Vlados021 for translating the problem.[/hide]

Find all functions $f:\mathbb Z_{>0}\to \mathbb Z_{>0}$ such that $a+f(b)$ divides $a^2+bf(a)$ for all positive integers $a$ and $b$ with $a+b>2019$.

Find all functions $f : \mathbb{R} \to \mathbb{Z}$ which satisfy the conditions: $f(x+y) < f(x) + f(y)$ $f(f(x)) = \lfloor {x} \rfloor + 2$

$AD$ is the angle bisector of the right triangle $ABC$ with $\angle ABC = 60^o$ and $\angle BCA = 90^o$. $E$ is chosen on $\overline{AB}$ so that the line parallel to $\overline{DE}$ through $C$ bisects $\overline{AE}$. Find $\angle EDB$ in degrees.

What is angle $B$ of triangle$ ABC$, if it is known that the altitudes drawn from $A$ and $C$ intersect inside the triangle and one of them is divided by of intersection point into equal parts, and the other one in the ratio of $2: 1$, counting from the vertex?

Points $L$ and $H$ are marked on the sides $AB$ of an acute-angled triangle ABC so that $CL$ is a bisector and $CH$ is an altitude. Let $P,Q$ be the feet of the perpendiculars from $L$ to $AC$ and $BC$ respectively. Prove that $AP \cdot BH = BQ \cdot AH$. I. Gorodnin

The [i]liar's guessing game[/i] is a game played between two players $A$ and $B$. The rules of the game depend on two positive integers $k$ and $n$ which are known to both players. At the start of the game $A$ chooses integers $x$ and $N$ with $1 \le x \le N.$ Player $A$ keeps $x$ secret, and truthfully tells $N$ to player $B$. Player $B$ now tries to obtain information about $x$ by asking player $A$ questions as follows: each question consists of $B$ specifying an arbitrary set $S$ of positive integers (possibly one specified in some previous question), and asking $A$ whether $x$ belongs to $S$. Player $B$ may ask as many questions as he wishes. After each question, player $A$ must immediately answer it with [i]yes[/i] or [i]no[/i], but is allowed to lie as many times as she wants; the only restriction is that, among any $k+1$ consecutive answers, at least one answer must be truthful. After $B$ has asked as many questions as he wants, he must specify a set $X$ of at most $n$ positive integers. If $x$ belongs to $X$, then $B$ wins; otherwise, he loses. Prove that: 1. If $n \ge 2^k,$ then $B$ can guarantee a win. 2. For all sufficiently large $k$, there exists an integer $n \ge (1.99)^k$ such that $B$ cannot guarantee a win. [i]Proposed by David Arthur, Canada[/i]

Let $P(x)$ be a polynomial with integer coefficients. We will denote the set of all prime numbers by $\mathbb P$. Show that the set $\mathbb S := \{p\in\mathbb P : \exists\text{ }n \text{ s.t. }p\mid P(n)\}$ is finite if and only if $P(x)$ is a non-zero constant polynomial.

Prove that the equation $x^2 + x + 1 = py$ has solution $(x,y)$ for the infinite number of simple $p$.

Sequence {$x_n$} satisfies: $x_1=1$ , ${x_n=\sqrt{x_{n-1}^2+x_{n-1}}+x_{n-1}}$ ( ${n>=2}$ ) Find the general term of {$x_n$}

The sides of triangle are $x$, $2x+1$ and $x+2$ for some positive rational $x$. Angle of triangle is $60$ degree. Find perimeter

Prove that none of the numbers $2^{2^n}+ 1$, $n = 0, 1, 2, \dots$ is a perfect cube.

For a given odd prime number $p$, define $f(n)$ the remainder of $d$ divided by $p$, where $d$ is the biggest divisor of $n$ which is not a multiple of $p$. For example when $p=5$, $f(6)=1, f(35)=2, f(75)=3$. Define the sequence $a_1, a_2, \ldots, a_n, \ldots$ of integers as the followings: [list] [*]$a_1=1$ [*]$a_{n+1}=a_n+(-1)^{f(n)+1}$ for all positive integers $n$. [/list] Determine all integers $m$, such that there exist infinitely many positive integers $k$ such that $m=a_k$.