RIOPLATENSE 2000

Let $a$ and $b$ be positive integers such that the number $b^2 + (b +1)^2 +...+ (b + a)^2-3$ is multiple of $5$ and $a + b$ is odd. Calculate the digit of the units of the number $a + b$ written in decimal notation.

In a triangle $ABC$ , points $D, E$ and $F$ are considered on the sides $BC, CA$ and $AB$ respectively, such that the areas of the triangles $AFE, BFD$ and $CDE$ are equal. Prove that $$\frac{(DEF) }{ (ABC)} \ge \frac{1}{4}$$ Note: $(XYZ)$ is the area of triangle $XYZ$ .

Let $n>1$ be an integer. For each numbers $(x_1, x_2,\dots, x_n)$ with $x_1^2+x_2^2+x_3^2+\dots +x_n^2=1$ , denote $m=\min\{|x_i-x_j|, 0<i<j<n+1\}$ Find the maximum value of $m$ .

Let $a, b$ and $c$ be positive integers such that $a^2 + b^2 + 1 = c^2$ . Prove that $[a/2] + [c / 2]$ is even.

Note: $[x]$ is the integer part of $x$ .

Let $ABC$ be a triangle with $AB < AC$ , let $L$ be midpoint of arc $BC$ (the point $A$ is not in this arc) of the circumcircle $w$ ( $ABC$ ). Let $E$ be a point in $AC$ where $AE = \frac{AB + AC}{2}$ , the line $EL$ intersects $w$ in $P$ . If $M$ and $N$ are the midpoints of $AB$ and $BC$ , respectively, prove that $AL, BP$ and $MN$ are concurrents

Let $g(x) = ax^2 + bx + c$ a quadratic function with real coefficients such that the equation $g(g(x)) = x$ has four distinct real roots. Prove that there isn't a function $f$ : $R--R$ such that $f(f(x)) = g(x)$ for all $x$ real