VMO 1999 — Division A

Determine all real solutions of the system
$$\begin{cases}(1+4^{2x-y})\,5^{y-2x+1}=2^{2x-y+1}+1,\\ y^3+4x+\ln(y^2+2x)+1=0.\end{cases}$$

Let $ABC$ be a triangle. Let $A'$ be the midpoint of the arc $BC$ of its circumcircle that does not contain $A$; define $B'$ and $C'$ similarly. The segments $A'B',B'C',C'A'$ meet the sides of $ABC$ at six points, two on each side, dividing each side into three parts. Prove that the three middle subsegments are equal if and only if $ABC$ is equilateral.

The sequence $a_1,a_2,a_3,\ldots$ is defined by $a_1=1$, $a_2=2$, and $a_{n+2}=3a_{n+1}-a_n$. The sequence $b_1,b_2,b_3,\ldots$ is defined by $b_1=1$, $b_2=4$, and $b_{n+2}=3b_{n+1}-b_n$. Prove that positive integers $a,b$ satisfy $5a^2-b^2=4$ if and only if $a=a_n$ and $b=b_n$ for some positive integer $n$.

Determine the greatest value of
$$\frac2{x^2+1}-\frac2{y^2+1}+\frac3{z^2+1}$$
for positive real numbers $x,y,z$ satisfying $xyz+x+z=y$.

Let $OA,OB,OC,OD$ be four rays in space such that the angle between any two is the same. Prove that, as a ray $OX$ varies, the sum of the cosines of $\angle XOA,\angle XOB,\angle XOC,\angle XOD$ is constant, and the sum of the squares of those cosines is also constant.

Determine all functions $f(n)$ defined on the nonnegative integers, with values in $\{0,1,2,\ldots,2000\}$, satisfying:
(a) $f(n)=n$ for $0\le n\le2000$;
(b) $f(f(m)+f(n))=f(m+n)$ for all positive integers $m,n$.