RIOPLATENSE 1993

Find all functions $f$ defined on the integers greater than or equal to $1$ that satisfy: (a) for all $n,f(n)$ is a positive integer. (b) $f(n + m) =f(n)f(m)$ for all $m$ and $n$ . (c) There exists $n_0$ such that $f(f(n_0)) = [f(n_0)]^2$ .

An integer is written in each cell of a board of $N$ rows and $N + 1$ columns. Prove that some columns (possibly none) can be deleted so that in each row the sum of the numbers left uncrossed out is even.

Given three points $A, B$ and $C$ (not collinear) construct the equilateral triangle of greater perimeter such that each of its sides passes through one of the given points.

$x$ and $y$ are real numbers such that $6 -x$ , $3 + y^2$ , $11 + x$ , $14 - y^2$ are greater than zero. Find the maximum of the function $$f(x,y) = \sqrt{(6 -x)(3 + y^2)} + \sqrt{(11 + x)(14 - y^2)}.$$

Prove that for every integer $k \ge 2$ there are $k$ different natural numbers $n_1$ , $n_2$ , $...$ , $n_k$ such that: $$\frac{1}{n_1}+\frac{1}{n_2}+...+\frac{1}{n_k}=\frac{3}{17}$$

Let $ABCDE$ be pentagon such that $AE = ED$ and $BC = CD$ . It is known that $\angle BAE + \angle EDC + \angle CB A = 360^o$ and that $P$ is the midpoint of $AB$ . Show that the triangle $ECP$ is right.