Consider two triangles, $ABC$ and $DEF$, and any point $O$. We take any point $X$ in $\vartriangle ABC$ and any point $Y$ in $\vartriangle DEF$ and draw a parallelogram $OXY Z$. Prove that the locus of all possible points $Z$ form a polygon. How many sides can it have? Prove that its perimeter is equal to the sum of perimeters of the original triangles.

Given a $m \times n$ rectangle where $m,n\geq 2023$. The square in the $i$-th row and $j$-th column is filled with the number $i+j$ for $1\leq i \leq m, 1\leq j \leq n$. In each move, Alice can pick a $2023 \times 2023$ subrectangle and add $1$ to each number in it. Alice wins if all the numbers are multiples of $2023$ after a finite number of moves. For which pairs $(m,n)$ can Alice win? [i]Proposed by Boon Qing Hong[/i]

Find all function $f:\mathbb{R}\rightarrow\mathbb{R}$ such that for all $x,y\in\mathbb{R}$ the following equality holds \[ f(\left\lfloor x\right\rfloor y)=f(x)\left\lfloor f(y)\right\rfloor \] where $\left\lfloor a\right\rfloor $ is greatest integer not greater than $a.$ [i]Proposed by Pierre Bornsztein, France[/i]

Three circles $\omega_1,\omega_2,\omega_3$ are tangent to line $l$ at points $A,B,C$ ($B$ lies between $A,C$) and $\omega_2$ is externally tangent to the other two. Let $X,Y$ be the intersection points of $\omega_2$ with the other common external tangent of $\omega_1,\omega_3$. The perpendicular line through $B$ to $l$ meets $\omega_2$ again at $Z$. Prove that the circle with diameter $AC$ touches $ZX,ZY$. [i]Proposed by Iman Maghsoudi - Siamak Ahmadpour[/i]

Let $P(x,\, y)=2x^{2}-6xy+5y^{2}$. Let us say an integer number $a$ is a value of $P$ if there exist integer numbers $b$, $c$ such that $P(b,\, c)=a$. a) Find all values of $P$ lying between 1 and 100. b) Show that if $r$ and $s$ are values of $P$, then so is $rs$.

Consider real numbers $x,y,z$ such that $x,y,z>0$. Prove that \[ (xy+yz+xz)\left(\frac{1}{x^2+y^2}+\frac{1}{x^2+z^2}+\frac{1}{y^2+z^2}\right) > \frac{5}{2}. \]

Let $a$,$b$,$c$ be positive real numbers satisfying $abc=1$. Prove that inequality $\dfrac{a^k}{a+b}+ \dfrac{b^k}{b+c}+\dfrac{c^k}{c+a}\ge \dfrac{3}{2}$ holds for all integer $k$ ($k \ge 2$).

Let $ABCD$ be a trapezoid with $AD$ parallel to $BC$, $AD = 2$, and $BC = 1$. Let $M$ be the midpoint of $AD$, and let $P$ be the intersection of $BD$ with $CM$. Extend $AP$ to meet segment $CD$ at point $Q$. If the ratio $CQ/QD = a/b$, where $a$ and $b$ are positive integers and $\text{gcd}(a, b) = 1$, find $a + b$.

Let $ABCD$ be a parallelogram. Let $X$ and $Y$ be arbitrary points on sides $BC$ and $CD$, respectively. Segments $BY$ and $DX$ intersect at $P$. Prove that the line going through the midpoints of segments $BD$ and $XY$ is either parallel to or coincides with line $AP$.

Let $ABC$ be an acute,non-isosceles triangle with $AB<AC<BC$.Let $D,E,Z$ be the midpoints of $BC,AC,AB$ respectively and segments $BK,CL$ are altitudes.In the extension of $DZ$ we take a point $M$ such that the parallel from $M$ to $KL$ crosses the extensions of $CA,BA,DE$ at $S,T,N$ respectively (we extend $CA$ to $A$-side and $BA$ to $A$-side and $DE$ to $E$-side).If the circumcirle $(c_{1})$ of $\triangle{MBD}$ crosses the line $DN$ at $R$ and the circumcirle $(c_{2})$ of $\triangle{NCD}$ crosses the line $DM$ at $P$ prove that $ST\parallel PR$.

Let $AD$ and $BE$ be the altitudes of acute triangle $ABC.$ The circles with diameters $AD$ and $BE$ intersect at points $S$ and $T$. Prove that $\angle ACS=\angle BCT.$

Find all pairs of positive integers $(a,b)$ such that $5a^b - b = 2004$.

Let $a_1,a_2,\ldots a_n,k$, and $M$ be positive integers such that $$\frac{1}{a_1}+\frac{1}{a_2}+\cdots+\frac{1}{a_n}=k\quad\text{and}\quad a_1a_2\cdots a_n=M.$$ If $M>1$, prove that the polynomial $$P(x)=M(x+1)^k-(x+a_1)(x+a_2)\cdots (x+a_n)$$ has no positive roots.

Given a right triangle $T$, where the coordinates of its vertices are integers, let $E$ be the number of points of integer coordinates that belong to the edge of the triangle $T$, $I$ the number of points of integer coordinates that belong to the interior of the triangle $T$. Show that the area $A(T)$ of triangle $T$ is given by: $A(T) = \frac{E}{2}+I -1$.

Let $ABC$ be a triangle. Draw circles $\omega_A$, $\omega_B$, and $\omega_C$ such that $\omega_A$ is tangent to $AB$ and $AC$, and $\omega_B$ and $\omega_C$ are defined similarly. Let $P_A$ be the insimilicenter of $\omega_B$ and $\omega_C$. Define $P_B$ and $P_C$ similarly. Prove that $AP_A$, $BP_B$, and $CP_C$ are concurrent. [i]Tom Lu.[/i]

Let $MN$ be a chord of a circle with diameter $AB$, and let $A'$ and $B'$ be the orthogonal projections of $A$ and $B$ onto $MN$. Prove that $MA'=B'N$.

Find x, y, z $\in Z$\\$x^2+y^2+z^2=2^n(x+y+z)$\\$n\in N$

For which integers $n\geq 2$, the number $(n-1)^{n^{n+1}}+(n+1)^{n^{n-1}}$ is divisible by $n^{n}$ ?

The excircle $\omega_B$ of triangle $ABC$ opposite $B$ touches side $AC$, rays $BA$ and $BC$ at $B_1, C_1$ and $A_1$, respectively. Point $D$ lies on major arc $A_1C_1$ of $\omega_B$. Rays $DA_1$ and $C_1B_1$ meet at $E$. Lines $AB_1$ and $BE$ meet at $F$. Prove that line $FD$ is tangent to $\omega_B$ (at $D$).

In the figure, $ABC$ and $DAE$ are isosceles triangles ($AB = AC = AD = DE$) and the angles $BAC$ and $ADE$ have measures $36^o$. a) Using geometric properties, calculate the measure of angle $\angle EDC$. b) Knowing that $BC = 2$, calculate the length of segment $DC$. c) Calculate the length of segment $AC$ . [img]https://1.bp.blogspot.com/-mv43_pSjBxE/XqBMTfNlRKI/AAAAAAAAL2c/5ILlM0n7A2IQleu9T4yNmIY_1DtrxzsJgCK4BGAYYCw/s400/2004%2Bobm%2Bl2.png[/img]

All vertices of a regular 2016-gon are initially white. What is the least number of them that can be painted black so that:\\ (a) There is no right triangle\\ (b) There is no acute triangle\\ having all vertices in the vertices of the 2016-gon that are still white?

Let $(b_{n})_{n\in \mathbb{N}}$ be a sequence of positive real numbers such that $b_{0}=1$, $b_{n}=2+\sqrt{b_{n-1}}-2\sqrt{1+\sqrt{b_{n-1}}}$. Calculate $$\sum_{n=1}^{\infty}b_{n}2^{n}.$$

There are 2020 inhabitants in a town. Before Christmas, they are all happy; but if an inhabitant does not receive any Christmas card from any other inhabitant, he or she will become sad. Unfortunately, there is only one post company which offers only one kind of service: before Christmas, each inhabitant may appoint two different other inhabitants, among which the company chooses one to whom to send a Christmas card on behalf of that inhabitant. It is known that the company makes the choices in such a way that as many inhabitants as possible will become sad. Find the least possible number of inhabitants who will become sad.

A single player game has the following rules: initially, there are $10$ piles of stones with $1,2,...,10$ stones, respectively. A movement consists on making one of the following operations: [b]i)[/b] to choose $2$ piles, both of them with at least $2$ stones, combine them and then add $2$ stones to the new pile; [b]ii)[/b] to choose a pile with at least $4$ stones, remove $2$ stones from it, and then split it into two piles with amount of piles to be chosen by the player. The game continues until is not possible to make an operation. Show that the number of piles with one stone in the end of the game is always the same, no matter how the movements are made.

A circle of radius 1 is tangent to a circle of radius 2. The sides of $ \triangle ABC$ are tangent to the circles as shown, and the sides $ \overline{AB}$ and $ \overline{AC}$ are congruent. What is the area of $ \triangle ABC$? [asy]defaultpen(black+linewidth(0.7)); size(7cm); real t=2^0.5; D((0,0)--(4*t,0)--(2*t,8)--cycle, black); D(CR((2*t,2),2), black); D(CR((2*t,5),1), black); dot(origin^^(4t,0)^^(2t,8)); label("B", (0,0), SW); label("C", (4*t,0), SE); label("A", (2*t,8), N); D((2*t,2)--(2*t,4), black); D((2*t,5)--(2*t,6), black); MP('2', (2*t,3), W); MP('1',(2*t, 5.5), W);[/asy] $ \textbf{(A) } \frac {35}2 \qquad \textbf{(B) } 15\sqrt {2} \qquad \textbf{(C) } \frac {64}3 \qquad \textbf{(D) } 16\sqrt {2} \qquad \textbf{(E) } 24$