Show that if $a,b,c$ are complex numbers that such that \[ (a+b)(a+c)=b \qquad (b+c)(b+a)=c \qquad (c+a)(c+b)=a\]then $a,b,c$ are real numbers.

a) Let $f,g:\mathbb{Z}\rightarrow\mathbb{Z}$ be one to one maps. Show that the function $h:\mathbb{Z}\rightarrow\mathbb{Z}$ defined by $h(x)=f(x)g(x)$, for all $x\in\mathbb{Z}$, cannot be a surjective function. b) Let $f:\mathbb{Z}\rightarrow\mathbb{Z}$ be a surjective function. Show that there exist surjective functions $g,h:\mathbb{Z}\rightarrow\mathbb{Z}$ such that $f(x)=g(x)h(x)$, for all $x\in\mathbb{Z}$.

The sides of a triangle have lengths $a,b,c$. Prove that: \begin{align*}(-a+b+c)(a-b+c)\, +\, & (a-b+c)(a+b-c)+(a+b-c)(-a+b+c)\\ &\le\sqrt{abc}(\sqrt{a}+\sqrt{b}+\sqrt{c})\end{align*}

Three schools have $200$ students each. Every student has at least one friend in each school (if the student $a$ is a friend of the student $b$ then $b$ is a friend of $a$). It is known that there exists a set $E$ of $300$ students (among the $600$) such that for any school $S$ and any two students $x,y\in E$ but not in $S$, the number of friends in $S$ of $x$ and $y$ are different. Show that one can find a student in each school such that they are friends with each other.

Find all polynomials with real coefficients $P$ such that \[ P(x)P(2x^2-1)=P(x^2)P(2x-1)\] for every $x\in\mathbb{R}$.

The vertices $A,B,C$ and $D$ of a square lie outside a circle centred at $M$. Let $AA',BB',CC',DD'$ be tangents to the circle. Assume that the segments $AA',BB',CC',DD'$ are the consecutive sides of a quadrilateral $p$ in which a circle is inscribed. Prove that $p$ has an axis of symmetry.

Find the least $n\in N$ such that among any $n$ rays in space sharing a common origin there exist two which form an acute angle.

Show that the set of positive integers that cannot be represented as a sum of distinct perfect squares is finite.

Let $n$ be a positive integer and $f(x)=a_mx^m+\ldots + a_1X+a_0$, with $m\ge 2$, a polynomial with integer coefficients such that: a) $a_2,a_3\ldots a_m$ are divisible by all prime factors of $n$, b) $a_1$ and $n$ are relatively prime. Prove that for any positive integer $k$, there exists a positive integer $c$, such that $f(c)$ is divisible by $n^k$.

Let $ p$ and $ q$ be relatively prime positive integers. A subset $ S$ of $ \{0, 1, 2, \ldots \}$ is called ideal if $ 0 \in S$ and for each element $ n \in S,$ the integers $ n + p$ and $ n + q$ belong to $ S.$ Determine the number of ideal subsets of $ \{0, 1, 2, \ldots \}.$

Find all pairs $\left(m,n\right)$ of positive integers, with $m,n\geq2$, such that $a^n-1$ is divisible by $m$ for each $a\in \left\{1,2,3,\ldots,n\right\}$.

Prove that there is no function $f:(0,\infty )\rightarrow (0,\infty)$ such that \[f(x+y)\ge f(x)+yf(f(x)) \] for every $x,y\in (0,\infty )$.

The tangents at $A$ and $B$ to the circumcircle of the acute triangle $ABC$ intersect the tangent at $C$ at the points $D$ and $E$, respectively. The line $AE$ intersects $BC$ at $P$ and the line $BD$ intersects $AC$ at $R$. Let $Q$ and $S$ be the midpoints of the segments $AP$ and $BR$ respectively. Prove that $\angle ABQ=\angle BAS$.

Consider a convex polyhedron $P$ with vertices $V_1,\ldots ,V_p$. The distinct vertices $V_i$ and $V_j$ are called neighbours if they belong to the same face of the polyhedron. To each vertex $V_k$ we assign a number $v_k(0)$, and construct inductively the sequence $v_k(n)\ (n\ge 0)$ as follows: $v_k(n+1)$ is the average of the $v_j(n)$ for all neighbours $V_j$ of $V_k$ . If all numbers $v_k(n)$ are integers, prove that there exists the positive integer $N$ such that all $v_k(n)$ are equal for $n\ge N$ .