Find all functions $ f: \mathbb R\longrightarrow \mathbb R$ such that for each $ x,y\in\mathbb R$: \[ f(xf(y)) + y + f(x) = f(x + f(y)) + yf(x)\]

Suppose that $ I$ is incenter of triangle $ ABC$ and $ l'$ is a line tangent to the incircle. Let $ l$ be another line such that intersects $ AB,AC,BC$ respectively at $ C',B',A'$. We draw a tangent from $ A'$ to the incircle other than $ BC$, and this line intersects with $ l'$ at $ A_1$. $ B_1,C_1$ are similarly defined. Prove that $ AA_1,BB_1,CC_1$ are concurrent.

Suppose that $ T$ is a tree with $ k$ edges. Prove that the $ k$-dimensional cube can be partitioned to graphs isomorphic to $ T$.

Let $ P_1,P_2,P_3,P_4$ be points on the unit sphere. Prove that $ \sum_{i\neq j}\frac1{|P_i-P_j|}$ takes its minimum value if and only if these four points are vertices of a regular pyramid.

Let $a,b,c > 0$ and $ab+bc+ca = 1$. Prove that: \[ \sqrt {a^3 + a} + \sqrt {b^3 + b} + \sqrt {c^3 + c}\geq2\sqrt {a + b + c}. \]

Prove that in a tournament with 799 teams, there exist 14 teams, that can be partitioned into groups in a way that all of the teams in the first group have won all of the teams in the second group.

Let $ S$ be a set with $ n$ elements, and $ F$ be a family of subsets of $ S$ with $ 2^{n-1}$ elements, such that for each $ A,B,C\in F$, $ A\cap B\cap C$ is not empty. Prove that the intersection of all of the elements of $ F$ is not empty.

Find all polynomials $ p$ of one variable with integer coefficients such that if $ a$ and $ b$ are natural numbers such that $ a + b$ is a perfect square, then $ p\left(a\right) + p\left(b\right)$ is also a perfect square.

$ I_a$ is the excenter of the triangle $ ABC$ with respect to $ A$, and $ AI_a$ intersects the circumcircle of $ ABC$ at $ T$. Let $ X$ be a point on $ TI_a$ such that $ XI_a^2=XA.XT$. Draw a perpendicular line from $ X$ to $ BC$ so that it intersects $ BC$ in $ A'$. Define $ B'$ and $ C'$ in the same way. Prove that $ AA'$, $ BB'$ and $ CC'$ are concurrent.

In the triangle $ ABC$, $ \angle B$ is greater than $ \angle C$. $ T$ is the midpoint of the arc $ BAC$ from the circumcircle of $ ABC$ and $ I$ is the incenter of $ ABC$. $ E$ is a point such that $ \angle AEI=90^\circ$ and $ AE\parallel BC$. $ TE$ intersects the circumcircle of $ ABC$ for the second time in $ P$. If $ \angle B=\angle IPB$, find the angle $ \angle A$.

$ k$ is a given natural number. Find all functions $ f: \mathbb{N}\rightarrow\mathbb{N}$ such that for each $ m,n\in\mathbb{N}$ the following holds: \[ f(m)+f(n)\mid (m+n)^k\]

In the acute-angled triangle $ ABC$, $ D$ is the intersection of the altitude passing through $ A$ with $ BC$ and $ I_a$ is the excenter of the triangle with respect to $ A$. $ K$ is a point on the extension of $ AB$ from $ B$, for which $ \angle AKI_a=90^\circ+\frac 34\angle C$. $ I_aK$ intersects the extension of $ AD$ at $ L$. Prove that $ DI_a$ bisects the angle $ \angle AI_aB$ iff $ AL=2R$. ($ R$ is the circumradius of $ ABC$)