Let $ P$ be a point in triangle $ ABC$, and define $ \alpha,\beta,\gamma$ as follows: \[ \alpha=\angle BPC-\angle BAC, \quad \beta=\angle CPA-\angle \angle CBA, \quad \gamma=\angle APB-\angle ACB.\] Prove that \[ PA\dfrac{\sin \angle BAC}{\sin \alpha}=PB\dfrac{\sin \angle CBA}{\sin \beta}=PC\dfrac{\sin \angle ACB}{\sin \gamma}.\]

Solve the equation \[ x+a^3=\sqrt[3]{a-x}\] where $ a$ is a real parameter.

Let $ a_1,a_2,a_3,\dots$ be infinite sequence of positive integers satisfying the following conditon: for each prime number $ p$, there are only finite number of positive integers $ i$ such that $ p|a_i$. Prove that that sequence contains a sub-sequence $ a_{i_1},a_{i_2},a_{i_3},\dots$, with $ 1 \le i_1<i_2<i_3<\dots$, such that for each $ m \ne n$, $ \gcd(a_{i_m},a_{i_n})=1$.

Let $ n$ and $ k$ be positive integers. Please, find an explicit formula for \[ \sum y_1y_2 \dots y_k,\] where the summation runs through all $ k-$tuples positive integers $ (y_1,y_2,\dots,y_k)$ satisfying $ y_1+y_2+\dots+y_k=n$.

Call an $n$-gon to be lattice if its vertices are lattice points. Prove that inside every lattice convex pentagon there exists a lattice point.

Let $a > 3$ be an odd integer. Show that for every positive integer $n$ the number $a^{2^n}- 1$ has at least $n + 1$ distinct prime divisors.

Let $a, b, c$ be positive reals such that $a + b + c = 1$ and $P(x) = 3^{2005}x^{2007 }- 3^{2005}x^{2006} - x^2$. Prove that $P(a) + P(b) + P(c) \le -1$.

Given a collection of sets $X = \{A_1, A_2, ..., A_n\}$. A set $\{a_1, a_2, ..., a_n\}$ is called a single representation of $X$ if $a_i \in A_i$ for all i. Let $|S| = mn$, $S = A_1\cup A_2 \cup ... \cup A_n = B_1 \cup B_2 \cup ... \cup B_n$ with $|A_i| = |B_i| = m$ for all $i$. Prove that $S = C_1 \cup C_2 \cup ... \cup C_n$ where for every $i, C_i $ is a single represenation for $\{A_j\}_{j=1}^n $and $\{B_j\}_{j=1}^n$.

Given triangle $ ABC$ and its circumcircle $ \Gamma$, let $ M$ and $ N$ be the midpoints of arcs $ BC$ (that does not contain $ A$) and $ CA$ (that does not contain $ B$), repsectively. Let $ X$ be a variable point on arc $ AB$ that does not contain $ C$. Let $ O_1$ and $ O_2$ be the incenter of triangle $ XAC$ and $ XBC$, respectively. Let the circumcircle of triangle $ XO_1O_2$ meets $ \Gamma$ at $ Q$. (a) Prove that $ QNO_1$ and $ QMO_2$ are similar. (b) Find the locus of $ Q$ as $ X$ varies.

Let $ a,b,c$ be non-zero real numbers satisfying \[ \dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}=\dfrac{1}{a+b+c}.\] Find all integers $ n$ such that \[ \dfrac{1}{a^n}+\dfrac{1}{b^n}+\dfrac{1}{c^n}=\dfrac{1}{a^n+b^n+c^n}.\]

For each real number $ x$< let $ \lfloor x \rfloor$ be the integer satisfying $ \lfloor x \rfloor \le x < \lfloor x \rfloor +1$ and let $ \{x\}=x-\lfloor x \rfloor$. Let $ c$ be a real number such that \[ \{n\sqrt{3}\}>\dfrac{c}{n\sqrt{3}}\] for all positive integers $ n$. Prove that $ c \le 1$.

Let $ X$ be a set of $ k$ vertexes on a plane such that no three of them are collinear. Let $ P$ be the family of all $ {k \choose 2}$ segments that connect each pair of points. Determine $ \tau(P)$.

Let $ a,b,c$ be real numbers. Prove that $ (ab+bc+ca-1)^2 \le (a^2+1)(b^2+1)(c^2+1)$.

Let $ ABCD$ be a convex quadrtilateral such that $ AB$ is not parallel with $ CD$. Let $ \Gamma_1$ be a circle that passes through $ A$ and $ B$ and is tangent to $ CD$ at $ P$. Also, let $ \Gamma_2$ be a circle that passes through $ C$ and $ D$ and is tangent to $ AB$ at $ Q$. Let the circles $ \Gamma_1$ and $ \Gamma_2$ intersect at $ E$ and $ F$. Prove that $ EF$ passes through the midpoint of $ PQ$ iff $ BC \parallel AD$.

Find all pairs of function $ f: \mathbb{N} \rightarrow \mathbb{N}$ and polynomial with integer coefficients $ p$ such that: (i) $ p(mn) = p(m)p(n)$ for all positive integers $ m,n > 1$ with $ \gcd(m,n) = 1$, and (ii) $ \sum_{d|n}f(d) = p(n)$ for all positive integers $ n$.

Let $ S$ be a finite family of squares on a plane such that every point on that plane is contained in at most $ k$ squares in $ S$. Prove that $ P$ can be divided into $ 4(k-1)+1$ sub-family such that in each sub-family, each pair of squares are disjoint.

Let $ ABCD$ be a cyclic quadrilateral and $ O$ be the intersection of diagonal $ AC$ and $ BD$. The circumcircles of triangle $ ABO$ and the triangle $ CDO$ intersect at $ K$. Let $ L$ be a point such that the triangle $ BLC$ is similar to $ AKD$ (in that order). Prove that if $ BLCK$ is a convex quadrilateral, then it has an incircle.

Find all functions $ f: \mathbb{R} \rightarrow \mathbb{R}$ satisfying \[ f(f(x + y)) = f(x + y) + f(x)f(y) - xy\]for all real numbers $x$ and $y$.

On each vertex of a regular $ n-$gon there was a crow. Call this as initial configuration. At a signal, they all flew by and after a while, those $ n$ crows came back to the $ n-$gon, one crow for each vertex. Call this as final configuration. Determine all $ n$ such that: there are always three crows such that the triangle they formed in the initial configuration and the triangle they formed in the final configuration are both right-angled triangle.

Determine all pairs $ (n,p)$ of positive integers, where $ p$ is prime, such that $ 3^p-np=n+p$.