In $\triangle AEF$, let $B$ and $D$ be on segments $AE$ and $AF$ respectively, and let $ED$ and $FB$ intersect at $C$. Define $K,L,M,N$ on segments $AB,BC,CD,DA$ such that $\frac{AK}{KB}=\frac{AD}{BC}$ and its cyclic equivalents. Let the incircle of $\triangle AEF$ touch $AE,AF$ at $S,T$ respectively; let the incircle of $\triangle CEF$ touch $CE,CF$ at $U,V$ respectively. Prove that $K,L,M,N$ concyclic implies $S,T,U,V$ concyclic.

Find all functions $f$ and $g$ defined from $\mathbb{R}_{>0}$ to $\mathbb{R}_{>0}$ such that for all $x, y > 0$ the two equations hold $$ (f(x) + y - 1)(g(y) + x - 1) = {(x + y)}^2 $$ $$ (-f(x) + y)(g(y) + x) = (x + y + 1)(y - x - 1) $$ [i]Note: $\mathbb{R}_{>0}$ denotes the set of positive real numbers.[/i]

Given a positive integer $n$, let $a_1,a_2,..,a_n$ be a sequence of nonnegative integers. A sequence of one or more consecutive terms of $a_1,a_2,..,a_n$ is called $dragon$ if their aritmetic mean is larger than 1. If a sequence is a $dragon$, then its first term is the $head$ and the last term is the $tail$. Suppose $a_1,a_2,..,a_n$ is the $head$ or/and $tail$ of some $dragon$ sequence; determine the minimum value of $a_1+a_2+\cdots +a_n$ in terms of $n$.

Let $ABC$ be a triangle, and let $D$ be the point where the incircle meets side $BC$. Let $J_b$ and $J_c$ be the incentres of the triangles $ABD$ and $ACD$, respectively. Prove that the circumcentre of the triangle $AJ_bJ_c$ lies on the angle bisector of $\angle BAC$.

A quadrilateral ABCD is inscribed into a circle ω with center O. The circumcircle of the triangle AOC intersects the lines AB, BC, CD and DA the second time at the points M, N, K and L respectively. Prove that the lines MN, KL and the tangents to ω at the points A и C all touch the same circle.

How many ordered pairs of integers $(x, y)$ satisfy the equation$$x^{2020}+y^2=2y?$$ $\textbf{(A) } 1 \qquad\textbf{(B) } 2 \qquad\textbf{(C) } 3 \qquad\textbf{(D) } 4 \qquad\textbf{(E) } \text{infinitely many}$

How many different positive integers divide $10!$ ?

At the beginning of the academic year, the Olympic Center must accept a certain number of talented students for the 2024 different classes it offers. Although the admitted students are given freedom of choice in classes, there are certain rules. So, any student must take at least one class and cannot take all the classes. At the same time, there cannot be a common class that all students take, and any class must be taken by at least one student. As a final addition to the center's rules, for any student and any class that this student did not enroll in (call this type of class A), the number of students in each A must be greater than the number of classes this student enrolled. At least how many students must the center accept for these rules to be valid?

The three points $A, B, C$ form a triangle. $AB=4, BC=5, AC=6$. Let the angle bisector of $\angle A$ intersect side $BC$ at $D$. Let the foot of the perpendicular from $B$ to the angle bisector of $\angle A$ be $E$. Let the line through $E$ parallel to $AC$ meet $BC$ at $F$. Compute $DF$.

Determine all functions $f:(0,\infty)\to\mathbb{R}$ satisfying $$\left(x+\frac{1}{x}\right)f(y)=f(xy)+f\left(\frac{y}{x}\right)$$ for all $x,y>0$.

Prove that for positive real numbers $a,b,c$ satisfying $abc=1$ the following inequality holds: $$ \frac{a}{b(1+c)} +\frac{b}{c(1+a)}+\frac{c}{a(1+b)} \ge \frac{3}{2} $$

A tennis player computes her win ratio by dividing the number of matches she has won by the total number of matches she has played. At the start of a weekend, her win ratio is exactly $.500$. During the weekend, she plays four matches, winning three and losing one. At the end of the weekend, her win ratio is greater than $.503$. What's the largest number of matches she could've won before the weekend began?

Find all pairs of positive integers $(x,y)$ with the following property: If $a,b$ are relative prime and positive divisors of $ x^3 + y^3$, then $a+b - 1$ is divisor of $x^3+y^3$. (Cyprus)

Find the number of functions $ A\stackrel{f}{\longrightarrow } A $ for which there exist two functions $ A\stackrel{g}{\longrightarrow } B\stackrel{h}{\longrightarrow } A $ having the properties that $ g\circ h =\text{id.} $ and $ h\circ g=f, $ where $ B $ and $ A $ are two finite sets.

A real number $\alpha \geq 0$ is given. Find the smallest $\lambda = \lambda (\alpha ) > 0$, such that for any complex numbers ${z_1},{z_2}$ and $0 \leq x \leq 1$, if $\left| {{z_1}} \right| \leq \alpha \left| {{z_1} - {z_2}} \right|$, then $\left| {{z_1} - x{z_2}} \right| \leq \lambda \left| {{z_1} - {z_2}} \right|$.

If $ x\neq1$, $ y\neq1$, $ x\neq y$ and \[ \frac{yz\minus{}x^{2}}{1\minus{}x}\equal{}\frac{xz\minus{}y^{2}}{1\minus{}y}\] show that both fractions are equal to $ x\plus{}y\plus{}z$.

Given a triangle $ABC$. An arbitrary point $P$ is chosen on the circumcircle of triangle $ABH$ ($H$ is the orthocenter of triangle $ABC$). Lines $AP$ and $BP$ meet the opposite sidelines of the triangle at points $A' $ and $B'$, respectively. Determine the locus of midpoints of segments $A'B'$.

Find all positive integers $ k$ for which equation $ n^m\minus{}m^n\equal{}k$ has solution in positive integers.

Show that $n!=a^{n-1}+b^{n-1}+c^{n-1}$ has only finitely many solutions in positive integers. [i]Proposed by Dorlir Ahmeti, Albania[/i]

Let $m_1, m_2, \cdots ,m_k$ be positive numbers with the sum $S$. Prove that \[\displaystyle\sum_{i=1}^k\left(m_i +\frac{1}{m_i}\right)^2 \ge k\left(\frac{k}{S}+\frac{S}{k}\right)^2\]

Give two statements: (1) $a,b,c$ are complex numbers, if $a^2+b^2>c^2$, then $a^2+b^2-c^2>0$. (2) $a,b,c$ are complex numbers, if $a^2+b^2-c^2>0$, then $a^2+b^2>c^2$. Then, which is true? $\text{(A)}$ (1) is correct, (2) is correct as well $\text{(B)}$ (1) is correct, (2) is incorrect $\text{(C)}$ (1) is incorrect, (2) is incorrect as well $\text{(D)}$ (1) is incorrect, (2) is correct

Find all functions $ f,g:\mathbb{Q}\longrightarrow\mathbb{Q} $ that verify the relations $$ \left\{\begin{matrix} f(g(x)+g(y))=f(g(x))+y \\ g(f(x)+f(y))=g(f(x))+y\end{matrix}\right. , $$ for all $ x,y\in\mathbb{Q} . $

A number is placed in each cell of an $n \times n$ board so that the following holds: (A) the cells on the boundary all contain 0; (B) other cells on the main diagonal are each1 greater than the mean of the numbers to the left and right; (C) other cells are the mean of the numbers to the left and right. Show that (B) and (C) remain true if ''left and right'' is replaced by ''above and below''.

Let be given an acute triangle $ABC$. Show that there exist unique points $A_1 \in BC$, $B_1 \in CA$, $C_1 \in AB$ such that each of these three points is the midpoint of the segment whose endpoints are the orthogonal projections of the other two points on the corresponding side. Prove that the triangle $A_1B_1C_1$ is similar to the triangle whose side lengths are the medians of $\triangle ABC$.

The two circles $\Gamma_1$ and $\Gamma_2$ intersect at $P$ and $Q$. The common tangent that's on the same side as $P$, intersects the circles at $A$ and $B$,respectively. Let $C$ be the second intersection with $\Gamma_2$ of the tangent to $\Gamma_1$ at $P$, and let $D$ be the second intersection with $\Gamma_1$ of the tangent to $\Gamma_2$ at $Q$. Let $E$ be the intersection of $AP$ and $BC$, and let $F$ be the intersection of $BP$ and $AD$. Let $M$ be the image of $P$ under point reflection with respect to the midpoint of $AB$. Prove that $AMBEQF$ is a cyclic hexagon.