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 $TC \cdot S_{(MBT)}=TB \cdot S_{(MCT)}$

Each $1 \times 1$ square of an $n \times n$ table contains a different number. The smallest number in each row is marked, and these marked numbers are in different columns. Then the smallest number in each column is marked, and these marked numbers are in different rows. Prove that the two sets of marked numbers are identical. (V Klepcyn)