Juan and Pedro play alternately on the given grid. Each one in turn traces $1$ to $5$ routes different from the ones outlined above, that join $A$ with $B$, moving only to the right and upwards on the grid lines. Juan starts playing. The one who traces a route that passes through $C$ or $D$ loses. Prove that one of them can win regardless of how the other plays.

In Hidro planet were living $2008^2$ hydras some time ago. One of them had 1 tentacle, other 2 and so on to the last with $2008^2$ tentacles. Let $t(H)$ be the number of tentacles of hydra $H$. The pairing of $H_1$ and $H_2$ (where $t(H_1) < t(H_2)$) is a new hydra with $t(H_2)-t(H_1)+8$ tentacles, in case of $t(H_1)=t(H_2)$ they die. An expedition found in Hidro the last hydra with 23 tentacles. That could be true ?

In each square of an $n \times n$ board with $n\ge 2$, an integer is written not null. Said board is called Inca if for each square, the number written on it is equal to the difference of the numbers written on two of its neighboring squares (with a common side). For what values of $n$, can you get Inca boards ?

Show that when a prime number is divided by $30$, the remainder is $1$ or a prime number. Shows that if it is divided by $60$ or $90$ the same thing does not happen.

Let $I$ be the incenter of an acute riangle $ABC$. Let $C_A(A, AI)$ be the circle with center $A$ and radius $AI$. Circles $C_B(B, BI)$, $C_C(C, CI) $ are defined in an analogous way. Let $X, Y, Z$ be the intersection points of $C_B$ with $C_C$, $C_C$ with $C_A$, $C_A$ with $C_B$ respectively (different than $I$) . Show that if the radius of the circle that passes through the points $X, Y, Z$ is equal to the radius of the circle that passes through points $A$, $B$ and $C$ then triangle $ABC$ is equilateral.

Determine the smallest value of $x^2 + y^2 + z^2$, where $x, y, z$ are real numbers, so that $x^3 + y^3 + z^3 -3xyz = 1.$

Determine all the functions $f : R \to R$ such that: $$x + f(xf(y)) = f(y) + yf(x)$$for all $x, y \in R$.

Prove that there are infinitely many positive integers $n$ such that $\frac{5^n-1}{n+2}$ is an integer.

Let $\omega_1$ and $\omega_2$ be circles that intersect at points $A$ and $B$ and let $O_1$ and $O_2$ be their respective centers. We take $M$ in $\omega_1$ and $N$ in $\omega_2$ on the same side as $B$ with respect to segment $O_1O_2$, such that $MO_1\parallel BO_2$ and $BO_1 \parallel NO_2$. Draw the tangents to $\omega_1$ and $\omega_2$ through $M$ and $N$ respectively, which intersect at $K$. Show that $A$, $B$ and $K$ are collinear.

Let $x_1, x_2, ..., x_n$ be positive reals. Prove that $$\sum_{k=1}^n \frac{x_k(2x_k - x_{k+1} - x_{k+2})}{x_{k+1} + x_{k+2}} \ge 0$$In the sum, cyclic indices have been taken, that is, $x_{n+1} = x_1$ and $x_{n+2} = x_2$.

Let $ABC$ be an isosceles triangle with base $BC$ and $\angle BAC = 20^o$. Let $D$ a point on side $AB$ such that $AD = BC$. Determine $\angle DCA$.

Find all the triples of prime numbers $(p, q, r)$ such that $$p | 2qr + r \,\,\,, \,\,\,q |2pr + p \,\,\, and \,\,\, r | 2pq + q.$$