Let $O$ be the circumcenter and $H$ the orthocenter of an acute triangle $ABC$. Show that there exist points $D$, $E$, and $F$ on sides $BC$, $CA$, and $AB$ respectively such that \[ OD + DH = OE + EH = OF + FH\] and the lines $AD$, $BE$, and $CF$ are concurrent.

Find all polynomials $p(x)$ satisfying $p(x^3+1)=p(x+1)^3$ for all $x$.

Let $ H$ be the orthocenter of an acute-angled triangle $ ABC$. The circle $ \Gamma_{A}$ centered at the midpoint of $ BC$ and passing through $ H$ intersects the sideline $ BC$ at points $ A_{1}$ and $ A_{2}$. Similarly, define the points $ B_{1}$, $ B_{2}$, $ C_{1}$ and $ C_{2}$. Prove that the six points $ A_{1}$, $ A_{2}$, $ B_{1}$, $ B_{2}$, $ C_{1}$ and $ C_{2}$ are concyclic. [i]Author: Andrey Gavrilyuk, Russia[/i]

Let $a,b,c$ be positive reals such that $abc=1$. Prove the inequality $$\frac{4a-1}{(2b+1)^2} + \frac{4b-1}{(2c+1)^2} + \frac{4c-1}{(2a+1)^2}\geqslant 1.$$

Let $A=\{-2562,-2561,...,2561,2562\}$. Prove that for any bijection (1-1, onto function) $f:A\to A$, $$\sum_{k=1}^{2562}\left\lvert f(k)-f(-k)\right\rvert\text{ is maximized if and only if } f(k)f(-k)<0\text{ for any } k=1,2,...,2562.$$

Let $ABC$ be an acute triangle and let $D$ be the foot of the perpendicular from $A$ to side $BC$. Let $P$ be a point on segment $AD$. Lines $BP$ and $CP$ intersect sides $AC$ and $AB$ at $E$ and $F$, respectively. Let $J$ and $K$ be the feet of the peroendiculars from $E$ and $F$ to $AD$, respectively. Show that $\frac{FK}{KD}=\frac{EJ}{JD}$.

Chords $A_1A_2, A_3A_4, A_5A_6$ of a circle $\Omega$ concur at point $O$. Let $B_i$ be the second common point of $\Omega$ and the circle with diameter $OA_i$ . Prove that chords $B_1B_2, B_3B_4, B_5B_6$ concur.

Find all functions $f:\mathbb{R}^{+}\to \mathbb{R}^{+}$ such that for all positive real numbers $x$ and $y$, the following equation holds: \[(x+y)f(f(x)y)=x^2f(f(x)+f(y)).\]

For a sequence $(a_{n})_{n\geq 1}$ of real numbers it is known that $a_{n}=a_{n-1}+a_{n+2}$ for $n\geq 2$. What is the largest number of its consecutive elements that can all be positive?

Shikaku and his son Shikamaru must climb a staircase that has $2022$ steps; the steps are listed $1$, $2$, $...$ , $2022$ and the floor is considered step $0$. This bores them both a lot, so so they decide to organize a game. They begin by tying a rope between them, so that At most they can be separated from each other by a distance of $7$ steps, that is, if they are in the steps $m$ and$ n$, then it must always be true that $|m-n| \le 7$. For the game they establish the following rules: a) They move alternately in turns. b) In his corresponding turn, the player must move to a higher step than in the one that (the same) was previously. c) If a player has just moved to the $n$-th step, then on the next turn the other player cannot be moved to any of the steps $n-1$, $n$ or $n + 1$, except when it is for reach the last step. d) Whoever reaches the last step (listed with $2022$) wins. Shikamaru is bored to start, so his father starts. Determine which of the two players has a winning strategy and describe it.

Two rational numbers \(\tfrac{m}{n}\) and \(\tfrac{n}{m}\) are written on a blackboard, where \(m\) and \(n\) are relatively prime positive integers. At any point, Evan may pick two of the numbers \(x\) and \(y\) written on the board and write either their arithmetic mean \(\tfrac{x+y}{2}\) or their harmonic mean \(\tfrac{2xy}{x+y}\) on the board as well. Find all pairs \((m,n)\) such that Evan can write 1 on the board in finitely many steps. [i]Proposed by Yannick Yao[/i]

In the kindergarten there is a big box with balls of three colors: red, blue and green, 100 balls in total. Once Pasha took out of the box 30 red, 10 blue, and 20 green balls and played with them. Then he lost five balls and returned the others back into the box. The next day, Sasha took out of the box 8 red, 18 blue, and 48 green balls. Is it possible to determine the color of at least one lost ball?

A set of points $P_i$ [i]covers[/i] a polygon if for every point in the polygon, a line can be drawn inside the polygon to at least one $P_i$. Points $A_1, A_2, \cdots, A_n$ in the plane form a $2007$-gon, not necessarily convex. Find the minimum value of $n$ such that for any such polygon, we can pick $n$ points inside it that cover the polygon.

Let $a_1$, $a_2$, $\cdots$ be a sequence such that $a_1=a_2=\frac 15$, and for $n \ge 3$, $$a_n=\frac{a_{n-1}+a_{n-2}}{1+a_{n-1}a_{n-2}}.$$ Find the smallest integer $n$ such that $a_n>1-5^{-2022}$.

Find the limit \[\lim_{n\to\infty} \frac{1}{n\ln n}\int_{\pi}^{(n+1)\pi} (\sin ^ 2 t)(\ln t)\ dt.\]

Prove that among any nine divisors of $30^{2010}$ there are two whose product is a perfect square.