Let $x, y, z$ be positive numbers such that $xyz \le 2$ and $\frac{1}{x^2}+ \frac{1}{y^2}+ \frac{1}{z^2}< k$, for some real $k \ge 2$. Find all values of $k$ such that the conditions above imply that there exist a triangle having the side-lengths $x, y, z$.

We call a permutation $\sigma$ of the first $n$ positive integers friendly if and only if the following conditions are fulfilled: (1) $\sigma(k + 1) \in \{2\sigma(k), 2\sigma(k) - 1, 2\sigma(k) - n, 2\sigma(k) - n - 1\}, \forall k \in \{1, 2, ..., n - 1\}$ (2) $\sigma(1) \in \{2 \sigma(n), 2\sigma(n) - 1, 2\sigma(n) - n, 2\sigma(n) - n - 1\}$. Find all positive integers $n$ for which there exists such a friendly permutation of the first $n$ positive integers.

Given are on a line three points $A, B, C$ such that $AB = 1$ and $BC = x$. Consider the circles $\Omega_a, \Omega_b$ and $\Omega_c$ which are tangent to the given line at the points $A, B, C$ respectively, and such that $\Omega_b$ is tangent externally with both $\Omega_a$ and $\Omega_c$ in points $M, N$ respectively. Find all values of the radius of the circle $\Omega_b$ for which the triangle $BMN$ is isosceles.

Given are six reals $a, b, c, x, y, z$ such that $(a + b + c)(x + y + z) = 3$ and $(a^2 + b^2 + c^2)(x^2 + y^2 + z^2) = 4$. Prove that $ax + by + cz \ge 0$.

Let $\{a_n\}_{n\ge 0}$ be a sequence of rational numbers given by $a_0 = a_1 = a_2 = a_3 = 1$ and for all $n \ge 4$ we have $a_{n-4}a_n = a_{n-3}a_{n-1} + a^2_{n-2}$. Prove that all the terms of the sequence are integers.

Prove that if two triangles are inscribed in the same circle, then their incircles are not strictly contained one into each other.

Determine all values of $a \in R$ such that there exists a function $f : [0, 1] \to R$ fulfilling the following inequality for all $x \ne y$: $$|f(x) - f(y)| \ge a.$$

Let $ABC$ be a triangle with altitudes $AD, BE, CF$. Choose the points $A_1, B_1, C_1$ on the lines $AD, BE, CF$ respectively, such that $$\frac{AA_1}{AD}= \frac{BB_1}{BE}= \frac{CC_1}{CF} = k.$$Find all values of $k$ such that the triangle $A_1B_1C_1$ is similar to the triangle $ABC$ for all triangles $ABC$.

Prove that for every positive integer $m$ there exists a positive integer N such that $S(2^n) > m$ for every positive integer $n > N$, where by $S(x)$ we denote the sum of digits of a positive integer $x$.

The real polynomial $f \in R[X]$ has an odd degree and it is given that $f$ is co-prime with $g(x) = x^2 - x - 1$ and $$f(x^2 - 1) = f(x)f(-x), \forall x \in R.$$Prove that $f$ has at least two complex non-real roots.

Given is a finite set of points $M$ and an equilateral triangle $\Delta$ in the plane. It is known that for any subset $M' \subset M$, which has no more than $9$ points, can be covered by two translations of the triangle $\Delta$. Prove that the entire set $M$ can be covered by two translations of $\Delta$.

In a country there are $100$ cities, some of which are connected by roads. For each four cities there are at least two roads between them. Also, there is no path that passes through each city exactly one time. Prove that one can choose two cities among those $100$, such that each of the $98$ remaining cities would be connected by a road with at least one of the two chosen cities.

For which positive integers $n \ge 4$ one can find n points in the plane, no three collinear, such that for each triangle formed with three of the $n$ points which are on the convex hull, exactly one of the $n - 3$ remaining points belongs to its interior.

Let S be the set of positive integers $n$ for which $\frac{3}{n}$ cannot be written as the sum of two rational numbers of the form $\frac{1}{k}$, where $k$ is a positive integer. Prove that $S$ cannot be written as the union of finitely many arithmetic progressions.

Let $n \ge 3$ and $\sigma \in S_n$ a permutation of the first $n$ positive integers. Prove that the numbers $\sigma (1), 2\sigma (2), 3\sigma(3), ... , n\sigma (n)$ cannot form an arithmetic, nor a geometric progression.

Determine the parity of the positive integer $N$, where $$N = \lfloor \frac{2002!}{2001 \cdot2003} \rfloor.$$

A triangle $ABC$ is located in a cartesian plane $\pi$ and has a perimeter of $3 + 2\sqrt3$. It is known that the triangle $ABC$ has the property that any triangle in the plane $\pi$, congruent with it, contains inside or on the boundary at least one lattice point (a point with both coordinates integers). Prove that the triangle $ABC$ is equilateral.

At a party there were some couples attending. As they arrive each person gets to talk with all the other persons which are already in the room. During the party, after all the guests arrive, groups of persons form, such that no two persons forming a couple belong to the same group, and for each two persons that do not form a couple, there is one and only one group to which both belong. Find the number of couples attending the party, knowing that there are less groups than persons at the party.

Fifty students take part in a mathematical competition where a set of $8$ problems is given (same set to each participant). The final result showed that a total of $171$ correct solutions were obtained. Prove that there are $3$ of the given problems that have been correctly solved by the same $3$ students.

Find all positive integers n with the property that $n^3 - 1$ is a perfect square.

A convex polygon $P$ can be partitioned into $27$ parallelograms. Prove that it can also be partitioned into $21$ parallelograms.