Determine the smallest integer $n \ge 2012$ for which it is possible to have $16$ consecutive integers on a $4 \times 4$ board so that, if we select $4$ elements of which there are not two in the same row or in the same column, the sum of them is always equal to $n$. For the $n$ found, show how to fill the board.

An equilateral triangle with side $3$ is divided into $9$ small equal equilateral triangles with sides of length $1$. Each vertex of a triangle small (bold dots) is numbered with a different number than the $1$ to $10$. Inside each small triangle, write the sum of the numbers corresponding to its three vertices. Prove that there are three small triangles for which it is verified that the sum of the numbers written inside is at least $48$.

Two players $A$ and $B$ take turns taking stones from a pile of $N$ stones. They play in the order $A$, $B$, $A$, $B$, $A$, $....$, $A$ starts the game and the one who takes out the last stone loses.$ B$ can serve on each play $1$, $2$ or 3 stones, while$ A$ can draw $2, 3, 4$ stones or $1$ stone in each turn f it is the last one in the pile. Determine for what values of $N$ does $A$ have a winning strategy, and for what values the winning strategy is $B$'s.

A subset of the set $\{1, 2, 3, ..., 30\}$ is called delicious if it doesn't contain elements a and b such that $a = 3b$. A delicious subset It is called super delicious if, in addition to being delicious, it is verified that no delicious subset has more elements than it has. Determine the number of super delicious subsets

Three players $A, B$ and $C$ take turns taking stones from a pile of $N$ stones. They play in the order $A$, $B$, $C$, $A$, $B$, $C$, $....$, $A$ starts the game and the one who takes the last stone loses. Players $A$ and $C$ They form a team against $B$, they agree on a strategy joint. $B$ can take $1, 2, 3, 4$ or $5$ stones on each move, while that $A$ and $C$ can each draw $1, 2$ or $3$ stones in each turn. Determine for which values of $N$ have winning strategies $A$ and $C$ , and for what values the winning strategy is $B$'s.

$2013$ people run a marathon on a straight road $4m$ wide broad. At any given moment, no two runners are closer $2$ m from each other. Prove that there are two runners that at that moment are more than $1052$ m from each other. Note: Consider runners as points.

Cris has the equation $-2x^2 + bx + c = 0$, and Cristian increases the coefficients of the Cris equation by $1$, obtaining the equation $-x^2 + (b + 1) x + (c + 1) = 0$. Mariloli notices that the real solutions of the Cristian's equation are the squares of the real solutions of the Cris equation. Find all possible values that can take the coefficients $b$ and $c$.

Two equal isosceles triangles $ABC$ and $ADB$, with $C$ and $D$ located in different halfplanes with respect to the line $AB$, share the base $AB$. The midpoints of $AC$ and $BC$ are denoted by $E$ and $F$ respectively. Show that $DE$ and $DF$ divide $AB$ into three equal parts length.

Find all the natural numbers that are $300$ times the sum of its digits.

We say that a positive integer is decomposed if it is prime and also If a line is drawn separating it into two numbers, those two numbers are never composite. For example 1997 is decomposed since it is prime, it is divided into: $1$, $997$; $19$, $97$; $199$, $7$ and none of those numbers are compound. How many decomposed numbers are there between $2000$ and $3000$?

Let the real numbers be $a, b, c, d$ with $a \ge b$ and $c \ge d$. Prove that the equation $$(x + a) (x + d) + (x + b) (x + c) = 0$$has real roots.

Let $ABC$ be a triangle with sides $BC = 13$, $CA = 14$ and $AB = 15$. We denote by $I$ the intersection point of the angle bisectors and $M$ to the midpoint of $AB$. The line $IM$ cuts at $P$ at the altitude drawn from $C$. Find the length of $CP$.

Let $x, y, z$ be positive real numbers whose sum is $2013$. Find the maximum possible value of $$\frac{(x^2+y^2+z^2)(x^3+y^3+z^3)}{ (x^4+y^4+z^4)}.$$

Prove that there are infinitely many pairs $(a, b)$ of positive integers with the following properties: $\bullet$ $a+b$ divides $ab+1$, $\bullet$ $a-b$ divides $ab -1$, $\bullet$ $b > 2$ and $a > b\sqrt3 - 1$.

Let ABC be a triangle with $\angle A = 90^o$, $\angle B = 75^o$, and $AB = 2$. Points $P$ and $Q$ of the sides $AC$ and $BC$ respectively, are such that $\angle APB = \angle CPQ$ and $\angle BQA = \angle CQP$. Calculate the lenght of $QA$.