Let $f:\mathbb{R}\longrightarrow \mathbb{R}$ be a function such that $f(x+y+xy)=f(x)+f(y)+f(xy)$ for all $x, y\in\mathbb{R}$. Prove that $f$ satisfies $f(x+y)=f(x)+f(y)$ for all $x, y\in\mathbb{R}$.

In the triangle $ABC $ the median $BD$ is drawn, which is divided into three equal parts by the points $E $ and $F$ ($BE = EF = FD$). It is known that $AD = AF$ and $AB = 1$. Find the length of the segment $CE$.

Let $m$ and $n$ be positive integers. Prove that if $m^{4^n+1} - 1$ is a prime number, then there exists an integer $t \ge 0$ such that $n = 2^t$.

Prove that for all non-negative real numbers $x,y,z$, not all equal to $0$, the following inequality holds $\displaystyle \dfrac{2x^2-x+y+z}{x+y^2+z^2}+\dfrac{2y^2+x-y+z}{x^2+y+z^2}+\dfrac{2z^2+x+y-z}{x^2+y^2+z}\geq 3.$ Determine all the triples $(x,y,z)$ for which the equality holds. [i]Milan Mitreski, Serbia[/i]

Two circles centers $O$ and $O'$, radii $R$ and $R'$, meet at two points. A variable line $L$ meets the circles at $A, C, B, D$ in that order and $\frac{AC}{AD} = \frac{CB}{BD}$. The perpendiculars from $O$ and $O'$ to $L$ have feet $H$ and $H'$. Find the locus of $H$ and $H'$. If $OO'^2 < R^2 + R'^2$, find a point $P$ on $L$ such that $PO + PO'$ has the smallest possible value. Show that this value does not depend on the position of $L$. Comment on the case $OO'^2 > R^2 + R'^2$.

Compute the remainder when $10^{2021}$ is divided by $10101$. [i]Proposed by Nathan Xiong[/i]

Let $t\in (1,2)$. Show that there exists a polynomial $P(x)=a_nx^n+a_{n-1}x^{n-1}+...+a_1x+a_0$ with the coefficients in $\{1,-1\}$ such that $\left|P(t)-2019\right| \leqslant 1.$ [i]Proposed by N. Safaei (Iran)[/i]

Solve the equation $x^2+y^4+1=6^z$ in the set of integers.

Two circles $U,V$ have distinct radii,tangent to each other externally at $T$.$A,B$ are points on $U,V$ respectively,both distinct from $T$,such that $\angle ATB=90$. (1)Prove that line $AB$ passes through a fixed point; (2)Find the locus of the midpoint of $AB$.

$13$ fractions are corrected by using each of the numbers $1,2,...,26$ once.[b]Example:[/b]$\frac{12}{5},\frac{18}{26}.... $ What is the maximum number of fractions which are integers?

$0<b<1,0<a<\frac{\pi}{4}$,$x=(\sin a)^{\log_{b}\sin a},y=(\cos a)^{\log_{b}\cos a},z=(\sin a)^{\log_{b}\cos a}$. Then the order of $x,y,z$ is $\text{(A)}x<z<y\qquad\text{(B)}y<z<x\qquad\text{(C)}z<x<y\qquad\text{(D)}x<y<z$

Prove that if $a > 0$, $b > 0$, $abc=1$, then $$a+b+c \ge 3$$

Let $ AB$ be a diameter of a circle $ S$, and let $ L$ be the tangent at $ A$. Furthermore, let $ c$ be a fixed, positive real, and consider all pairs of points $ X$ and $ Y$ lying on $ L$, on opposite sides of $ A$, such that $ |AX|\cdot |AY| \equal{} c$. The lines $ BX$ and $ BY$ intersect $ S$ at points $ P$ and $ Q$, respectively. Show that all the lines $ PQ$ pass through a common point.

Let $a_1, a_2, \ldots, a_n$ be real numbers, but not all of them null. Show that the equation $$\sqrt{x+a_1}+\sqrt{x+a_2}+\ldots+\sqrt{x+a_n}=n\sqrt{x}$$ has at most one real solution.

Let $S$ denote a square of side-length $7$, and let eight squares with side-length $3$ be given. Show that it is impossible to cover $S$ by those eight small squares with the condition: an arbitrary side of those (eight) squares is either coincided, parallel, or perpendicular to others of $S$ .

Find all functions from positive integers to itself such that $f(a+b)=f(a)+f(b)+f(c)+f(d)$ for all $c^2+d^2=2ab$

Let $f_1$, $f_2$, $f_3$, ..., be a sequence of numbers such that \[ f_n = f_{n - 1} + f_{n - 2} \] for every integer $n \ge 3$. If $f_7 = 83$, what is the sum of the first 10 terms of the sequence?

Let $n \geq 2$ be a positive integer. A subset of positive integers $S$ is said to be [i]comprehensive[/i] if for every integer $0 \leq x < n$, there is a subset of $S$ whose sum has remainder $x$ when divided by $n$. Note that the empty set has sum 0. Show that if a set $S$ is comprehensive, then there is some (not necessarily proper) subset of $S$ with at most $n-1$ elements which is also comprehensive.

For all integers $n\ge 2$ with the following property: [list] [*] for each pair of positive divisors $k,~\ell <n$, at least one of the numbers $2k-\ell$ and $2\ell-k$ is a (not necessarily positive) divisor of $n$ as well.[/list]

Let $f$ of degree at most 13 such that $f(k) = 13^k$ for $0 \leq k \leq 13$. Compute the last three digits of $f(14)$. [i]Proposed by Kaylee Ji[/i]

Two real number sequences are guiven, one arithmetic $\left(a_n\right)_{n\in \mathbb {N}}$ and another geometric sequence $\left(g_n\right)_{n\in \mathbb {N}}$ none of them constant. Those sequences verifies $a_1=g_1\neq 0$, $a_2=g_2$ and $a_{10}=g_3$. Find with proof that, for every positive integer $p$, there is a positive integer $m$, such that $g_p=a_m$.

Let $A,B$ be two matrices with positive integer entries such that sum of entries of a row in $A$ is equal to sum of entries of the same row in $B$ and sum of entries of a column in $A$ is equal to sum of entries of the same column in $B$. Show that there exists a sequence of matrices $A_1,A_2,A_3,\cdots , A_n$ such that all entries of the matrix $A_i$ are positive integers and in the sequence \[A=A_0,A_1,A_2,A_3,\cdots , A_n=B,\] for each index $i$, there exist indexes $k,j,m,n$ such that \[\begin{array}{*{20}{c}} \\ {{A_{i + 1}} - {A_{i}} = } \end{array}\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} \quad \quad \ \ j& \ \ \ {k} \end{array}} \\ {\begin{array}{*{20}{c}} m \\ n \end{array}\left( {\begin{array}{*{20}{c}} { + 1}&{ - 1} \\ { - 1}&{ + 1} \end{array}} \right)} \end{array} \ \text{or} \ \begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} \quad \quad \ \ j& \ \ \ {k} \end{array}} \\ {\begin{array}{*{20}{c}} m \\ n \end{array}\left( {\begin{array}{*{20}{c}} { - 1}&{ + 1} \\ { + 1}&{ - 1} \end{array}} \right)} \end{array}.\] That is, all indices of ${A_{i + 1}} - {A_{i}}$ are zero, except the indices $(m,j), (m,k), (n,j)$, and $(n,k)$.

Let $ABC$ be an isosceles triangle with $\measuredangle C=\measuredangle B=36$. The point $M$ is in interior of $ ABC$ such that $\measuredangle MBC=24^{\circ} , \measuredangle BCM=30^{\circ}$ $N = AM \cap BC.$. Find $\measuredangle MCB$ .

Solve in none-negative integers ${x^3} + 7{x^2} + 35x + 27 = {y^3}$.

Given are two points $B,C$. Consider point $A$ not lying on the line $BC$ and draw the circles $C_1(K_1,R_1)$ (with center $K_1$ and radius $R_1$) and $C_2(K_2,R_2)$ with chord $AB, AC$ respectively such that their centers lie on the interior of the triangle $ABC$ and also $R_1 \cdot AC= R_2 \cdot AB$. Let $T$ be the intersection point of the two circles, different from $A$, and M be a random pointof line $AT$, prove that $TC \cdot S_{(MBT)}=TB \cdot S_{(MCT)}$