Let $\vartriangle ABC$ be an acute triangle. $R$ is a point on arc $BC$. Choose two points $P, Q$ on $AR$ such that $B,P,C,Q$ are concyclic. Let the second intersection of $BP$, $CP$, $BQ$, $CQ$ and the circumcircle of $\vartriangle ABC$ is $P_B$, $P_C$, $Q_B$, $Q_C$, respectively. Let the circumcenter of $\vartriangle P P_BP_C$ and $\vartriangle QQ_BQ_C$ are $O_P$ and $O_Q$, respectively. Prove that $A,O_P,O_Q,R$ are concylic. [i]proposed by andychang[/i]

The [i]traveler ant[/i] is walking over several chess boards. He only walks vertically and horizontally through the squares of the boards and does not pass two or more times over the same square of a board. a) In a $4$x$4$ board, from which squares can he begin his travel so that he can pass through all the squares of the board? b) In a $5$x$5$ board, from which squares can he begin his travel so that he can pass through all the squares of the board? c) In a $n$x$n$ board, from which squares can he begin his travel so that he can pass through all the squares of the board?

Find the value of $x^2+y^2+z^2$ where $x, y, z$ are non-zero and satisfy the following: (1) $x+y+z=3$ (2) $x^2(\frac{1}{y}+\frac{1}{z})+y^2(\frac{1}{z}+\frac{1}{x})+z^2(\frac{1}{x}+\frac{1}{y})=-3$

A [i]figuric [/i] is a convex polyhedron with $26^{5^{2019}}$ faces. On every face of a figuric we write down a number. When we throw two figurics (who don't necessarily have the same set of numbers on their sides) into the air, the figuric which falls on a side with the greater number wins; if this number is equal for both figurics, we repeat this process until we obtain a winner. Assume that a figuric has an equal probability of falling on any face. We say that one figuric rules over another if when throwing these figurics into the air, it has a strictly greater probability to win than the other figuric (it can be possible that given two figurics, no figuric rules over the other). Milisav and Milojka both have a blank figuric. Milisav writes some (not necessarily distinct) positive integers on the faces of his figuric so that they sum up to $27^{5^{2019}}$. After this, Milojka also writes positive integers on the faces of her figuric so that they sum up to $27^{5^{2019}}$. Is it always possible for Milojka to create a figuric that rules over Milisav's? [i]Proposed by Bojan Basic[/i]

Suppose that the sum of the squares of two complex numbers $x$ and $y$ is 7 and the sum of the cubes is 10. What is the largest real value that $x + y$ can have?

We say distinct positive integers  $a_1,a_2,\ldots ,a_n $ are "good" if their sum is equal to the sum of all pairwise $\gcd $'s among them. Prove that there are infinitely many $n$ s such that $n$ good numbers exist. [i]Proposed by Morteza Saghafian[/i]

Determine all positive integers n such that $n^2/ (n - 1)!$

Given a natural number $n$, prove that $2^{2n}-1$ is a multiple of $3$.

A line in the plane of a triangle $T$ is called an [i]equalizer[/i] if it divides $T$ into two regions having equal area and equal perimeter. Find positive integers $a>b>c,$ with $a$ as small as possible, such that there exists a triangle with side lengths $a,b,c$ that has exactly two distinct equalizers.

For a $ n\times n $ real matrix $ M, $ prove that [b]a)[/b] $ M=0 $ if $ \text{tr} \left(M^tM\right) =0. $ [b]b)[/b] $ ^tM=M $ if $M^tM=M^2. $ [b]c)[/b] $ ^tM=-M $ if $ M^tM=-M^2. $ [b]d)[/b] Give example of a $ 2\times 2 $ real matrix $ A $ satisfying the following: $ \text{(i)} ^tA\cdot A^2=A^3 $ and $ ^tA\neq A $ $ \text{(ii)} ^tA\cdot A^2=-A^3 $ and $ ^tA\neq -A $ [i]Vasile Pop[/i]

In a lake there are several sorts of fish, in the following distribution: $ 18\%$ catfish, $ 2\%$ sturgeon and $ 80\%$ other. Of a catch of ten fishes, let $ x$ denote the number of the catfish and $ y$ that of the sturgeons. Find the expectation of $ \frac {x}{y \plus{} 1}$

Let $A$ be a set of $65$ integers with pairwise different remainders modulo $2016$. Prove that exists a subset $B=\{a,b,c,d\}$ of set $A$ such that $a+b-c-d$ is divisible with $2016$

Let $a$ be a positive integer. We say that a positive integer $b$ is [i]$a$-good[/i] if $\tbinom{an}{b}-1$ is divisible by $an+1$ for all positive integers $n$ with $an \geq b$. Suppose $b$ is a positive integer such that $b$ is $a$-good, but $b+2$ is not $a$-good. Prove that $b+1$ is prime.

Find a positive integral solution to the equation \[\frac{1+3+5+\dots+(2n-1)}{2+4+6+\dots+2n}=\frac{115}{116}\] $\textbf{(A) }110\qquad\textbf{(B) }115\qquad\textbf{(C) }116\qquad\textbf{(D) }231\qquad\textbf{(E) }\text{The equation has no positive integral solutions.}$

Emerald writes $ 2009^2$ integers in a $ 2009\times 2009$ table, one number in each entry of the table. She sums all the numbers in each row and in each column, obtaining $ 4018$ sums. She notices that all sums are distinct. Is it possible that all such sums are perfect squares?

There are $n$ knights numbered $1$ to $n$ and a round table with $n$ chairs. The first knight chooses his chair, and from him, the knight number $k+1$ sits $ k$ places to the right of knight number $k$ , for all $1 \le k\le n-1$ (occupied and empty seats are counted). In particular, the second knight sits next to the first. Find all values ​​of $n$ such that the $n$ gentlemen occupy the $n$ chairs following the described procedure.

Each day $289$ students are divided into $17$ groups of $17$. No two students are ever in the same group more than once. What is the largest number of days that this can be done?

The triangle $ABC$ has $\angle BCA=90^{\circ}.$ Bisector of angle $\angle CAB$ intersects the side $BC$ in point $P$ and bisector of angle $\angle ABC$ intersects the side $AC$ in point $Q.$ If $M$ and $N$ are projections of $P$ and $Q$ on side $AB$, find the measure of the angle $\angle MCN.$

Alphonse and Beryl are playing a game. The game starts with two rectangles with integer side lengths. The players alternate turns, with Alphonse going first. On their turn, a player chooses one rectangle, and makes a cut parallel to a side, cutting the rectangle into two pieces, each of which has integer side lengths. The player then discards one of the three rectangles (either the one they did not cut, or one of the two pieces they cut) leaving two rectangles for the other player. A player loses if they cannot cut a rectangle. Determine who wins each of the following games: (a) The starting rectangles are $1 \times 2020$ and $2 \times 4040$. (b) The starting rectangles are $100 \times 100$ and $100 \times 500$.

A rectangular sheet of paper with integer length sides is given. The sheet is marked with unit squares. Arrows are drawn at each lattice point on the sheet in a way that each arrow is parallel to one of its sides, and the arrows at the boundary of the paper do not point outwards. Prove that there exists at least one pair of neighboring lattice points (horizontally, vertically or diagonally) such that the arrows drawn at these points are in opposite directions.

Find all functions $f:\mathbb{N}\rightarrow\mathbb{N}$ such that $$n!+f(m)!|f(n)!+f(m!)$$ for all $m,n\in\mathbb{N}$ [i]Proposed by Valmir Krasniqi and Dorlir Ahmeti, Albania[/i]

If point $M(x,y)$ lies on the line with equation $y=x+2$ and $1<y<3$, calculate the value of $A=\sqrt{y^2-8x}+\sqrt{y^2+2x+5}$

Let $f:\mathbb{Q}\times\mathbb{Q}\to\mathbb{Q}$ be a function satisfying: [list] [*] For any $x_1,x_2,y_1,y_2 \in \mathbb Q$, $$f\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right) \leq \frac{f(x_1,y_1)+f(x_2,y_2)}{2}.$$ [*] $f(0,0) \leq 0$. [*] For any $x,y \in \mathbb Q$ satisfying $x^2+y^2>100$, the inequality $f(x,y)>1$ holds.\ Prove that there is some positive rational number $b$ such that for all rationals $x,y$, $$f(x,y) \ge b\sqrt{x^2+y^2} - \frac{1}{b}.$$

Let $$F(x)=\frac{x^4}{\exp(x^3)}\int^x_0\int^{x-u}_0\exp(u^3+v^3)dvdu.$$Find $\lim_{x\to\infty}F(x)$ or prove that it does not exist.

Let $n$ be a positive integer. Every square in a $n \times n$-square grid is either white or black. How many such colourings exist, if every $2 \times 2$-square consists of exactly two white and two black squares? The squares in the grid are identified as e.g. in a chessboard, so in general colourings obtained from each other by rotation are different.