Consider a sequence of positive real numbers $ \left( x_n \right)_{n\ge 1} $ and a primitivable function $ f:\mathbb{R}\longrightarrow\mathbb{R} . $ [b]a)[/b] Prove that $ f $ is monotonic and continuous if for any natural numbers $ n $ and real numbers $ x, $ the inequality $$ f\left( x+x_n \right)\geqslant f(x) $$ is true. [b]b)[/b] Show that $ f $ is convex if for any natural numbers $ n $ and real numbers $ x, $ the inequality $$ f\left( x+2x_n \right) +f(x)\geqslant 2f\left( x+x_n \right) $$ is true. [i]Sorin Rădulescu[/i] and [i]Ion Savu[/i]

Let $f$ and $g$ be linear functions such that $f(g(2021))-g(f(2021)) = 20$. Compute $f(g(2022))- g(f(2022))$. (Note: A function h is linear if $h(x) = ax + b$ for all real numbers $x$.)

The centre $O$ of the circumcircle of $\vartriangle ABC$ lies inside the triangle. Perpendiculars are drawn rom $O$ on the sides. When produced beyond the sides they meet the circumcircle at points $K, M$ and $P$. Prove that $\overrightarrow{OK} + \overrightarrow{OM} + \overrightarrow{OP} = \overrightarrow{OI}$, where $I$ is the centre of the inscribed circle of $\vartriangle ABC$. (V Galperin, Moscow)

Let $m$ be a number containing only $0$ and $6$ as its digits.Show that $m$ can't be a perfect square.

The cyclic quadrilateral $ABCD$, inscribed in the circle $\Gamma$, satisfies $AB=BC$ and $CD=DA$, and $E$ is the intersection point of the diagonals $AC$ and $BD$. The circle with center $A$ and radius $AE$ intersects $\Gamma$ in two points $F$ and $G$. Prove that the line $FG$ is tangent to the circles with diameters $BE$ and $DE$.

Prove that for each positive integer $k$ there exists a number base $b$ along with $k$ triples of Fibonacci numbers $(F_u,F_v,F_w)$ such that when they are written in base $b$, their concatenation is also a Fibonacci number written in base $b$. (Fibonacci numbers are defined by $F_1 = F_2 = 1$ and $F_{n+2} = F_{n+1} + F_n$ for all positive integers $n$.) [i]Proposed by Serbia[/i]

Let $a, b,c$ be positive real numbers such that $ab + bc + ca = 1$. Prove that $$a\sqrt{b^2 + c^2 + bc} + b\sqrt{c^2 + a^2 + ca} + c\sqrt{a^2 + b^2 + ab} \ge \sqrt3$$

Two players take turns placing an unused number from {1, 2, 3, 4, 5, 6, 7, 8} into one of the empty squares in the array to the right. The game ends once all the squares are filled. The first player wins if the product of the numbers in the top row is greater. The second player wins if the product of the numbers in the bottom row is greater. If both players play with perfect strategy, who wins this game? [asy] unitsize(32); int[][] a = { {1, 2, 3, 4}, {5, 6, 7, 8}}; for (int i = 0; i < 4; ++i) { for (int j = 0; j < 2; ++j) { draw((i, -j)--(i+1, -j)--(i+1, -j-1)--(i, -j-1)--cycle); if (a[j][i] > 0) label(string(a[j][i]), (i+0.5, -j-0.5), fontsize(16pt)); } } [/asy]

A square $ABCD$ is given. A line passing through $A$ intersects $CD$ at $Q$. Draw a line parallel to $AQ$ that intersects the boundary of the square at points $M$ and $N$ such that the area of the quadrilateral $AMNQ$ is maximal.

Is it possible to fill a $3 \times 3$ grid with each of the numbers $1,2,\ldots,9$ once each such that the sum of any two numbers sharing a side is prime?

In a square $ABCD$, let $P$ and $Q$ be points on the sides $BC$ and $CD$ respectively, different from its endpoints, such that $BP=CQ$. Consider points $X$ and $Y$ such that $X\neq Y$, in the segments $AP$ and $AQ$ respectively. Show that, for every $X$ and $Y$ chosen, there exists a triangle whose sides have lengths $BX$, $XY$ and $DY$.

What is the least positive integer $k$ such that, in every convex $101$-gon, the sum of any $k$ diagonals is greater than or equal to the sum of the remaining diagonals?

Prove that for any n natural, the number \[ \sum \limits_{k=0}^{n} \binom{2n+1}{2k+1} 2^{3k} \] cannot be divided by $5$.

$ABC$ is an acute triangle. Points $M$ and $K$ on side $AC$ are such that $\angle ABM = \angle CBK$. Prove that the circumcenters of triangles $ABM$, $ABK$, $CBM$ and $CBK$ are concyclic. (Author: T. Emelyanova)

Let $ABC$ be an acute triangle with altitudes $AA'$ and $BB'$ and orthocenter $H$. Let $C_0$ be the midpoint of the segment $AB$. Let $g$ be the line symmetric to the line $CC_0$ with respect to the angular bisector of $\angle ACB$. Let $h$ be the line symmetric to the line $HC_0$ with respect to the angular bisector of $\angle AHB$. Show that the lines $g$ and $h$ intersect on the line $A'B'$.

A positive integer $n$ is fixed. Numbers $0$ and $1$ are placed in all cells (exactly one number in any cell) of a $k \times n$ table ($k$ is a number of the rows in the table, $n$ is the number of the columns in it). We call a table nice if the following property is fulfilled: for any partition of the set of the rows of the table into two nonempty subsets $R$[size=75]1[/size] and $R$[size=75]2[/size] there exists a nonempty set $S$ of the columns such that on the intersection of any row from $R$[size=75]1[/size] with the columns from $S$ there are even number of $1's$ while on the intersection of any row from $R$[size=75]2[/size] with the columns from $S$ there are odd number of $1's$. Find the greatest number of $k$ such that there exists at least one nice $k \times n$ table.

In a regular polygon $H$ of $6n+1$ sides ($n$ is a positive integer), we paint $r$ vertices red, and the rest blue. Demonstrate that the number of isosceles triangles that have three of their vertices of the same color does not depend on the way we distribute the colors on the vertices of $H$.

Esmeralda chooses two distinct positive integers \(a\) and \(b\), with \(b > a\), and writes the equation \[ x^2 - ax + b = 0 \] on the board. If the equation has distinct positive integer roots \(c\) and \(d\), with \(d > c\), she writes the equation \[ x^2 - cx + d = 0 \] on the board. She repeats the procedure as long as she obtains distinct positive integer roots. If she writes an equation for which this does not occur, she stops. a) Show that Esmeralda can choose \(a\) and \(b\) such that she will write exactly 2024 equations on the board. b) What is the maximum number of equations she can write knowing that one of the initially chosen numbers is 2024?

Prove that for any integer $n$, there exists a unique polynomial $Q$ with coefficients in $\{0,1,\ldots,9\}$ such that $Q(-2) = Q(-5) = n$.

Prove that there is no positive integer $n>1$ such that $n\mid2^{n} -1.$

Determine all positive integers $N$ for which the sphere \[ x^2+y^2+z^2=N \] has an inscribed regular tetrahedron whose vertices have integer coordinates.

Let $p=101.$ The sum \[\sum_{k=1}^{10}\frac1{\binom pk}\] can be written as a fraction of the form $\dfrac a{p!},$ where $a$ is a positive integer. Compute $a\pmod p.$

There are $2015$ distinct circles in a plane, with radius $1$. Prove that you can select $27$ circles, which form a set $C$, which satisfy the following. For two arbitrary circles in $C$, they intersect with each other or For two arbitrary circles in $C$, they don't intersect with each other.

In a scalene triangle $ABC,$ $I$ is the incenter and $O$ is the circumcenter. The line $IO$ intersects the lines $BC,CA,AB$ at points $D,E,F$ respectively. Let $A_1$ be the intersection of $BE$ and $CF$. The points $B_1$ and $C_1$ are defined similarly. The incircle of $ABC$ is tangent to sides $BC,CA,AB$ at points $X,Y,Z$ respectively. Let the lines $XA_1, YB_1$ and $ZC_1$ intersect $IO$ at points $A_2,B_2,C_2$ respectively. Prove that the circles with diameters $AA_2,BB_2$ and $CC_2$ have a common point.

$ \frac{\sqrt{10+\sqrt{1}}+\sqrt{10+\sqrt{2}}+...+\sqrt{10+\sqrt{99}}}{\sqrt{10-\sqrt{1}}+\sqrt{10-\sqrt{2}}+...+\sqrt{10-\sqrt{99}}}=? $