It is given isosceles triangle $ABC$ with $AB = AC$. $AD$ is diameter of circumcircle of triangle $ABC$. On the side $BC$ is chosen point $E$. On the sides $AC, AB$ there are points $F, G$ respectively such that $AFEG$ is parallelogram. Prove that $DE$ is perpendicular to $FG$.

A natural number $n\ge5$ leaves the remainder $2$ when divided by $3$. Prove that the square of $n$ is not a sum of a prime number and a perfect square.

Consider the integers formed using the digits $0,1,2,3,4,5,6$, without repetition. Find the largest multiple of $55$. Justify your answer.

Let $e_1, e_2, . . . e_{2019}$ be independently chosen from the set $\{0, 1, . . . , 20\}$ uniformly at random. Let $\omega = e^{\frac{2\pi}{i} 2019}$. Determine the expected value of $$|e_1\omega + e_2\omega^2 + ... + e_{2019}\omega^{2019}|.$$

In the space is given a tetrahedron with length of the edge $2$. Prove that distances from some point $M$ to all of the vertices of the tetrahedron are integer numbers if and only if $M$ is a vertex of tetrahedron. [i]J. Tabov[/i]

A quadratic trinomial $P(x)$ with the $x^2$ coefficient of one is such, that $P(x)$ and $P(P(P(x)))$ share a root. Prove that $P(0)*P(1)=0$.

If a right triangle can be covered by two unit circles, find the maximal area of the right triangle.

Compute the sum of roots of $(2 - x)^{2005} + x^{2005} = 0$.

Let $n,k$ be positive integers so that $n \ge k$.Find the maximum number of binary sequances of length $n$ so that fixing any arbitary $k$ bits they do not produce all binary sequances of length $k$.For exmple if $k=1$ we can only have one sequance otherwise they will differ in at least one bit which means that bit produces all binary sequances of length $1$.

In an acute-angled triangle $ABC$ on the side $AC$, point $P$ is chosen in such a way that $2AP = BC$. Points $X$ and $Y$ are symmetric to $P$ with respect to vertices $A$ and $C$, respectively. It turned out that $BX = BY$. Find $\angle BCA$.

In the coordinate space, each of the eight vertices of a rectangular box has integer coordinates. If the volume of the solid is $2011$, prove that the sides of the rectangular box are parallel to the coordinate axes.

Let $S$ be a set of $N \ge 3$ points in the plane. Assume that no $3$ points in $S$ are collinear. The segments with both endpoints in $S$ are colored in two colors. Prove that there is a set of $N - 1$ segments of the same color which don't intersect except in their endpoints such that no subset of them forms a polygon with positive area.

Suppose we have a sequence $a_1, a2_, ...$ of positive real numbers so that for each positive integer $n$, we have that $\sum_{k=1}^{n} a_ka_{\lfloor \sqrt{k} \rfloor} = n^2$. Determine the first value of $k$ so $a_k > 100$.

On the Cartesian plane, all points with both integer coordinates are painted blue. Two blue points are said to be [i]mutually visible[/i] if the line segment that connects them has no other blue points . Prove that there is a set of $ 2019$ blue points that are mutually visible two by two. [hide=official wording]No plano cartesiano, todos os pontos com ambas coordenadas inteiras são pintados de azul. Dois pontos azuis são ditos mutuamente visíveis se o segmento de reta que os conecta não possui outros pontos azuis. Prove que existe um conjunto de 2019 pontos azuis que são mutuamente visíveis dois a dois.[/hide] PS. There is a comment about problem being wrong / incorrect [url=https://artofproblemsolving.com/community/c6h1957974p14780265]here[/url]

If $a<b<c<d<e$ are consecutive positive integers such that $b+c+d$ is a perfect square and $a+b+c+d+e$ is a perfect cube, what is the smallest possible value of $c$?

Ninety-one students in a school are distributed in three classes. Each student took part in a competition. It is known that among any six students of the same sex some two got the same number of points. Show that here are four students of the same sex who are in the same class and who got the same number of points.

From the $9 \times 9$ chessboard, all $16$ unit squares whose row numbers and column numbers are both even have been removed. Disect the punctured board into rectangular pieces, with as few of them being unit squares as possible.

If $a$ and $b$ are the roots of $x^2 - 2x + 5$, what is $|a^8 + b^8|$?

Let $(x_n)$ be an infinite sequence of real numbers from interval $(0, 1)$. An infinite sequence $(a_n)$ of positive integers is defined as follows: $a_1 = 1$, and for $i \ge 1$, $a_{i+1}$ is equal to the smallest positive integer $m$, for which $[x_1 + x_2 + \ldots + x_m] = a_i$. Show that for any indexes $i, j$ holds $a_{i+j} \ge a_i + a_j$. [i]Proposed by Nazar Serdyuk[/i]

Let $p$ and $q$ be two different prime numbers. Prove that there are two positive integers, $a$ and $b$, such that the arithmetic mean of the divisors of $n=p^aq^b$ is an integer.

Several points are denoted on a white piece of paper. The distance between each two of the points is greater than $24$. A drop of ink was sprinkled over the paper covering an area smaller than $\pi$. Prove that there exists a vector $\overrightarrow v$ with $\overrightarrow v<1$, such that after translating all of the points by $v$ none of them is covered in ink.

How many $4-$digit numbers $\overline{abcd}$ are there such that $a<b<c<d$ and $b-a<c-b<d-c$ ?

Let $n\geq 3$ be a natural number and $x\in \mathbb{R}$, for which $\{ x\} =\{ x^2\} =\{ x^n\} $ (with $\{ x\} $ we denote the fractional part of $x$). Prove that $x$ is an integer.

$a,b,c,d$ are positive reals with sum $1$. Show that $\frac{a^2}{a+b}+\frac{b^2}{b+c}+\frac{c^2}{c+d}+\frac{d^2}{d+a} \ge \frac{1}{2}$ with equality iff $a=b=c=d=\frac{1}{4}$.

Find all pairs of real numbers $(x,y)$ satisfying the system of equations: \begin{align*}\frac{1}{x} + \frac{1}{2y} &= (x^2+3y^2)(3x^2+y^2)\\ \frac{1}{x} - \frac{1}{2y} &= 2(y^4-x^4)\end{align*}