Find all prime numbers $a$ and $b$ such that $$ 20a^3-b^3=1. $$

An $n\times n$ square table is divided into $n^2$ unit cells. Some unit segments of the obtained grid (i.e. the side of any unit cell) is colored black so that any unit cell of the given square has exactly one black side. Find a) the smallest b) the greatest possible number of black unit segments.

Let $H$ be the orthocenter of an acute triangle $ABC$, $AH=2$, $BH=12$, $CH=9$. Find the area of the triangle $ABC$.

Let four parallel lines $l_1$, $l_2$, $l_3$, and $l_4$ meet the hyperbola $y=1/x$ at points $A_1$ and $B_1$, $A_2$ and $B_2$, $A_3$ and $B_3$, $A_4$ and $B_4$, respectively. Prove that the areas of the quadrilaterals $A_1A_2A_3A_4$ and $B_1B_2B_3B_4$ are equal.

Let A and B be two disjoint subsets of positive integers such that |Al=n, lBl= m. It is known that for any k Є AUB at least one of the two following conditions holds 1) k + 17 Є A 2) k -31 Є B Find all possible values of m/n.

Find all positive numbers $a$, $b$, $c$, $d$ such that $a+b+c+d=1$ and $$ \max\left\{\frac{a^2}{b},\frac{b^2}{a}\right\}\cdot \min\left\{\frac{c^2}{d},\frac{d^2}{c}\right\}=(\min\{a+b,c+d\})^4. $$

Let $1=d_1<d_2<\ldots<d_k=n$ be all natural divisors of a natural number $n$. Find all possible values of $k$ if $n=d_2d_3+d_2d_5+d_3d_5$.

Given triangle $ABC$, let $D$ be an inner point of the side $BC$. Let $P$ and $Q$ be distinct inner points of the segment $AD$. Let $K=BP\cap AC$, $L=CP\cap AB$, $E=BQ\cap AC$, $F=CQ\cap AB$. Given that $KL\parallel EF$, find all possible values of the ratio $BD:DC$.

Let $I$ be the incenter of a non-isosceles triangle $ABC$. The line $AI$ intersects the circumcircle of the triangle $ABC$ at $A$ and $D$. Let $M$ be the middle point of the arc $BAC$. The line through the point $I$ perpendicular to $AD$ intersects $BC$ at $F$. The line $MI$ intersects the circle $BIC$ at $N$. Prove that the line $FN$ is tangent to the circle $BIC$.

Find all functions $f:\mathbb R\to\mathbb R$ such that $$ 4xyf(x^2-y^2)=(x^2-y^2)f(2x)f(2y) $$for all real $x$ and $y$.

The following operations are performed to the natural numbers $x$: $x\to x+2$ or $x\to x+n$, $x\to x\cdot 2$ or $x\to x\cdot n$. Additions and multiplications are performes alternatively (adding $2$ or $n$ and multiplying by $2$ or by $n$ one can choose as he wishes for each step). The number $m$ is called attainable if it can be obtained from $1$ by a sequence of such operations, otherwise $m$ is called unattainable. Prove that if $n=5$ or $n=7$, then there are infinitely many unattainable numbers.

On the side $ AB$ of a cyclic quadrilateral $ ABCD$ there is a point $ X$ such that diagonal $ BD$ bisects $ CX$ and diagonal $ AC$ bisects $ DX$. What is the minimum possible value of $ AB\over CD$? Proposed by S. Berlov

Given that $x$, $y$, $z$ are positive real numbers satiafying $x+y+z=2$, prove the inequality $$ \frac{(x-1)^2}{y}+\frac{(y-1)^2}{z}+\frac{(z-1)^2}{x}\geqslant\frac14\left(\frac{x^2+y^2}{x+y}+\frac{y^2+z^2}{y+z}+\frac{z^2+x^2}{z+x} \right). $$

Prove that for any positive integers $a$ and $b$ there exist infinitely many prime numbers $p$ such that $ap+b$ is a composite number. (Using Dirichlet's theorem is not allowed.)

Let $X$ be a finite set. Suppose that $X=A_1\sqcup\ldots\sqcup A_{10}$ and $X=B_1\sqcup\ldots\sqcup B_{10}$ are partitions of $X$ satisfying the following condition: if $A_i\cap B_j=\varnothing$ for some $1\leqslant i\leqslant j\leqslant 10$, then $|A_i\cup B_j|\geqslant 10$. Determine the smallest possible number of the elements in $X$.

Let $ABC$ be a triangle with circumcircle $\Gamma$ and incenter $I$ and let $M$ be the midpoint of $\overline{BC}$. The points $D$, $E$, $F$ are selected on sides $\overline{BC}$, $\overline{CA}$, $\overline{AB}$ such that $\overline{ID} \perp \overline{BC}$, $\overline{IE}\perp \overline{AI}$, and $\overline{IF}\perp \overline{AI}$. Suppose that the circumcircle of $\triangle AEF$ intersects $\Gamma$ at a point $X$ other than $A$. Prove that lines $XD$ and $AM$ meet on $\Gamma$. Proposed by Evan Chen, Taiwan

Let $a$, $b$, $c$ be the lengths of the sides of a triangle $ABC$. Prove that $$ a^2(p-a)(p-b)+b^2(p-b)(p-c)+c^2(p-c)(p-a)\leqslant\frac{4}{27}p^4, $$where $p$ is the half-perimeter of the triangle $ABC$.

For any positive integer $k$, denote the sum of digits of $k$ in its decimal representation by $S(k)$. Find all polynomials $P(x)$ with integer coefficients such that for any positive integer $n \geq 2016$, the integer $P(n)$ is positive and $$S(P(n)) = P(S(n)).$$ Proposed by Warut Suksompong, Thailand

Any cell of a $5\times 5$ table is colored black or white. Find the greatest possible value of the ways to place a $T$-tetramino on the table so that it covers exactly two white and two black cells (for various colorings of the table).

Given an isosceles triangle $ABC$ with $AB=AC$. let $\omega(XYZ)$ be the circumcircle of a triangle $XYZ$. Tangents to $\omega(ABC)$ at $B$ and $C$ meet at $D$. Point $F$ is marked on the arc $AB$ (opposite to $C$). Let $K$, $L$ be the intersection points of $AF$ and $BD$, $AB$ and $CF$, respectively. Prove that if circles $\omega(BTS)$ and $\omega(CFK)$ are tangent to each other, the their tangency point belongs to $AB$. (Here $T$ and $S$ are the centers of the circles $\omega(BLC)$ and $\omega(BLK)$, respectively.)

Given positive real $a$, $b$, $c$ such that $2abc+ab+bc+ca=1$, prove the inequality $$ \frac{a+1}{(a+1)^2+(b+1)^2}+\frac{b+1}{(b+1)^2+(c+1)^2}+\frac{c+1}{(c+1)^2+(a+1)^2}\leqslant 1. $$

Let $\tau(n)$ be the number of positive divisors of $n$. Let $\tau_1(n)$ be the number of positive divisors of $n$ which have remainders $1$ when divided by $3$. Find all positive integral values of the fraction $\frac{\tau(10n)}{\tau_1(10n)}$.

Let $n$ be a positive integer relatively prime to $6$. We paint the vertices of a regular $n$-gon with three colours so that there is an odd number of vertices of each colour. Show that there exists an isosceles triangle whose three vertices are of different colours.

Find the smallest constant $C > 0$ for which the following statement holds: among any five positive real numbers $a_1,a_2,a_3,a_4,a_5$ (not necessarily distinct), one can always choose distinct subscripts $i,j,k,l$ such that \[ \left| \frac{a_i}{a_j} - \frac {a_k}{a_l} \right| \le C. \]

Let $ABC$ be a triangle with $AB = AC \neq BC$ and let $I$ be its incentre. The line $BI$ meets $AC$ at $D$, and the line through $D$ perpendicular to $AC$ meets $AI$ at $E$. Prove that the reflection of $I$ in $AC$ lies on the circumcircle of triangle $BDE$.

Denote by $\mathbb{N}$ the set of all positive integers. Find all functions $f:\mathbb{N}\rightarrow \mathbb{N}$ such that for all positive integers $m$ and $n$, the integer $f(m)+f(n)-mn$ is nonzero and divides $mf(m)+nf(n)$. Proposed by Dorlir Ahmeti, Albania