RIOPLATENSE 2007

Determine the values of $n \in N$ such that a square of side $n$ can be split into a square of side $1$ and five rectangles whose side measures are $10$ distinct natural numbers and all greater than $1$ .

Let $ABC$ be a triangle with incenter $I$ . The circle of center $I$ which passes through $B$ intersects $AC$ at points $E$ and $F$ , with $E$ and $F$ between $A$ and $C$ and different from each other. The circle circumscribed to triangle $IEF$ intersects segments $EB$ and $FB$ at $Q$ and $R$ , respectively. Line $QR$ intersects the sides $A B$ and $B C$ at $P$ and $S$ , respectively. If $a , b$ and $c$ are the measures of the sides $B C, CA$ and $A B$ , respectively, calculate the measurements of $B P$ and $B S$ .

Let $p > 3$ be a prime number and $x$ an integer, denote by $r ( x )\in \{ 0 , 1 , ... , p - 1 \}$ to the rest of $x$ modulo $p$ . Let $x_1, x_2, ... , x_k$ ( $2 < k < p$ ) different integers modulo $p$ and not divisible by $p$ . We say that a number $a \in \{ 1 , 2 ,..., p -1 \}$ is good if $r ( a x_1) < r ( a x_2) <...< r ( a x_k)$ . Show that there are at most $\frac{2 p}{k + 1}-{ 1}$ good numbers.

Find all functions $f:Z\to Z$ with the following property: if $x+y+z=0$ , then $f(x)+f(y)+f(z)=xyz.$

Divide each side of a triangle into $50$ equal parts, and each point of the division is joined to the opposite vertex by a segment. Calculate the number of intersection points determined by these segments.

Clarification : the vertices of the original triangle are not considered points of intersection or division.

Let $n > 2$ be a natural number. A subset $A$ of $R$ is said $n$ -smallif there exist $n$ real numbers $t_1 , t_2 , ..., t_n$ such that the sets $t_1 + A , t_2 + A ,... , t_n + A$ are different . Show that $R$ can not be represented as a union of $n - 1$ $n$ -small sets .

Notation : if $r \in R$ and $B \subset R$ , then $r + B = \{ r + b | b \in B\}$ .