RIOPLATENSE 2001

Find all integer numbers $a, b, m$ and $n$ , such that the following two equalities are verified: $a^2+b^2=5mn$ and $m^2+n^2=5ab$

Let $ABC$ be an acute triangle and $A_1, B_1$ and $C_1$ , points on the sides $BC, CA$ and $AB$ , respectively, such that $CB_1 = A_1B_1$ and $BC_1 = A_1C_1$ . Let $D$ be the symmetric of $A_1$ with respect to $B_1C_1, O$ and $O_1$ are the circumcenters of triangles $ABC$ and $A_1B_1C_1$ , respectively. If $A \ne D, O \ne O_1$ and $AD$ is perpendicular to $OO_1$ , prove that $AB = AC$ .

For every integer $n > 1$ , the sequence $\left( {{S}_{n}} \right)$ is defined by ${{S}_{n}}=\left\lfloor {{2}^{n}}\underbrace{\sqrt{2+\sqrt{2+...+\sqrt{2}}}}_{n\ radicals} \right\rfloor$ where $\left\lfloor x \right\rfloor$ denotes the floor function of $x$ . Prove that ${{S}_{2001}}=2\,{{S}_{2000}}+1$ . .

Find all functions $f: R \to R$ such that, for any $x, y \in R$ : $f\left( f\left( x \right)-y \right)\cdot f\left( x+f\left( y \right) \right)={{x}^{2}}-{{y}^{2}}$

Let $ABC$ be a acute-angled triangle with centroid $G$ , the angle bisector of $\angle ABC$ intersects $AC$ in $D$ . Let $P$ and $Q$ be points in $BD$ where $\angle PBA = \angle PAB$ and $\angle QBC = \angle QCB$ . Let $M$ be the midpoint of $QP$ , let $N$ be a point in the line $GM$ such that $GN = 2GM$ (where $G$ is the segment $MN$ ), prove that: $\angle ANC + \angle ABC = 180$

For $m = 1, 2, 3, ...$ denote $S(m)$ the sum of the digits of $m$ , and let $f(m)=m+S(m)$ . Show that for each positive integer $n$ , there exists a number that appears exactly $n$ times in the sequence $f(1),f(2),...,f(m),...$