Pieces are placed in some squares of an $8 \times 8$ board sothat: a) There is at least one token in any rectangle with sides $2 \times 1$ or $1\times 2$. b) There are at least two neighboring pieces in any rectangle with sides $7\times 1$ or $1\times 7$. Find the smallest number of tokens that can be taken to fulfill with both conditions.

A prism is called binary if it can be assigned to each of its vertices a number from the set $\{-1, 1\}$, such that the product of the numbers assigned to the vertices of each face is equal to $-1$. a) Prove that the number of vertices of the binary prisms is divisible for $8$. b) Prove that a prism with $2000$ vertices is binary.

A tennis competition takes place over four days, the number of participants is $2n$ with $n \ge 5$. Each participant plays exactly once a day (a couple of participants may be more times). Prove that such competition can end with exactly one winner and exactly three players in second place and such that there are no players with four lost games,

Find all the real numbers $x, y$ such that $x^3 - y^3 = 7(x - y)$ and $x^3 + y^3 = 5(x + y).$

Find three different positive integers whose sum is minimum than meet the condition that the sum of each pair of them is a perfect square.

Let $ABCD$ be a quadrilateral that can be inscribed in a circle whose diagonals are perpendicular. Denote by $P$ and $Q$ the feet of the perpendiculars through $D$ and $C$ respectively on the line $AB$, $X$ is the intersection point of the lines $AC$ and $DP$, $Y$ is the intersection point of the lines $BD$ and $CQ$. Show that $XY CD$ is a rhombus.

Find all functions $f : R_+ \to R_+$ such that $$x^2(f(x)+f(y)) = (x+y)f(f(x)y)$$for all positive real $x, y$.

Prove that there is a unique positive integer formed only by the digits $2$ and $5$, which has $ 2007$ digits and is divisible by $2^{2007}$.

Let the triangle $ABC$ be acute. Let us take in the segment $BC$ two points $F$ and $G$ such that $BG > BF = GC$ and an interior point$ P$ to the triangle on the bisector of $\angle BAC$. Then are drawn through $P$, $PD\parallel AB$ and $PE \parallel AC$, $D \in AC$ and $E \in AB$, $\angle FEP = \angle PDG$. prove that $\vartriangle ABC$ is isosceles.

Prove that given $n$ points in the plane, not all aligned, there exists a line that passes through exactly two of them. original wordingDemostrar que dados n puntos en el plano, no todos alineados, existe una recta que pasa por exactamente dos de ellos.

For each positive integer $n$, let $S(n)$ be the sum of the digits of $n^2 +1$. A sequence $\{a_n\}$ is defined, with $a_0$ an arbitrary positive integer and $a_{n+1} = S(a_n)$. Prove that the sequence $\{a_n\}$ is eventually periodic with period three.

Let $O$ be the circumcircle of $\triangle ABC$, with $AC=BC$ end let $D=AO\cap BC$. If $BD$ and $CD$ are integer numbers and $AO-CD$ is prime, determine such three numbers.