Determine the smallest real number $a$ such that there is a square of side $a$ such that contains $5$ unit circles inside it without common interior points in pairs.

There are $n$ light bulbs in a circle and one of them is marked. Let operation $A$: Take a positive divisor $d$ of the number $n,$ starting with the light bulb marked and clockwise, we count around the circumference from $1$ to $dn$, changing the state (on or off) to those light bulbs that correspond to the multiples of $d$. Let operation $B$ be: Apply operation$ A$ to all positive divisors of $n$ (to the first divider that is applied is with all the light bulbs off and the remaining divisors is with the state resulting from the previous divisor). Determine all the positive integers $n$, such that when applying the operation on $B$, all the light bulbs are on.

There are two piles of cards, one with $n$ cards and the other with $m$ cards. $A$ and $B$ play alternately, performing one of the following actions in each turn. following operations: a) Remove a card from a pile. b) Remove one card from each pile. c) Move a card from one pile to the other. Player $A$ always starts the game and whoever takes the last one letter wins . Determine if there is a winning strategy based on $m$ and $n$, so that one of the players following her can win always.

Determine all the quadrilaterals that can be divided by a diagonal into two triangles of equal area and equal perimeter.

Determine the quadratic functions $f(x) = ax^2 + bx + c$ for which there exists an interval $(h, k)$ such that for all $x \in (h, k)$ it holds that $f(x)f(x + 1) < 0$ and $f(x)f(x -1) < 0$.

Determine all the quadruples of real numbers that satisfy the following: The product of any three of these numbers plus the fourth is constant.

Determine all functions $f : R_+ \to R$ such that:$$f(x)f(y) = f(xy) + \frac{1}{x} + \frac{1}{y}$$for all $x, y$ positive reals.

On the circumcircle of triangle $ABC$, point $P$ is taken in such a way that the perpendicular drawn by the point $P$ to the line $AC$ cuts the circle also at the point $Q$, the perpendicular drawn by the point $Q$ to the line $AB$ cuts the circle also at point R and the perpendicular drawn by point $R$ to the line BC cuts the circle also at the point $P$. Let $O$ be the center of this circle. Prove that $\angle POC = 90^o$ .

All positive differences $a_i -a_j$ of five different positive integers $a_1$, $a_2$, $a_3$, $a_4$ and $a_5$ are all different. Let $A$ be the set formed by the largest elements of each group of $5$ elements that meet said condition. Determine the minimum element of $A$.

Determine all triples of positive integers $(x, y, z)$ that satisfy $$x < y < z, \ \ gcd(x, y) = 6, \ \ gcd(y, z) = 10, \ \ gcd(z, x) = 8 \ \ and \ \ lcm(x, y, z) = 2400.$$

Find the smallest real number $A$, such that there are two different triangles, with integer sidelengths and so that the area of each be $A$.

Let $x_1, x_2, …, x_n$ and $y_1, y_2, …,y_n$ be positive reals such that $$x_1 + x_2 +.. + x_n \ge y_i \ge x^2_i$$for all $i = 1, 2, …, n$. Prove that $$\frac{x_1}{x_1y_1 + x_2}+ + \frac{x_2}{x_2y_2 + x_3} + ...+ \frac{x_n}{x_ny_n + x_1}> \frac{1}{2n}.$$