ÖMO 1985

Determine all quadruples $(a,b,c,d)$ of nonnegative integers satisfying:

$a^2+b^2+c^2+d^2=a^2 b^2 c^2$ .

For $n \in \mathbb{N}$ , let $f(n)=1^n+2^{n-1}+3^{n-2}+...+n^1$ . Determine the minimum value of: $\frac{f(n+1)}{f(n)}.$

A line meets the lines containing sides $BC,CA,AB$ of a triangle $ABC$ at $A_1,B_1,C_1,$ respectively. Points $A_2,B_2,C_2$ are symmetric to $A_1,B_1,C_1$ with respect to the midpoints of $BC,CA,AB,$ respectively. Prove that $A_2,B_2,$ and $C_2$ are collinear.

Find all natural numbers $n$ such that the equation:

$a_{n+1} x^2-2x \sqrt{a_1^2+a_2^2+...+a_{n+1}^2}+a_1+a_2+...+a_n=0$

has real solutions for all real numbers $a_1,a_2,...,a_{n+1}$ .

A sequence $(a_n)$ of positive integers satisfies: $a_n=\sqrt{\frac{a_{n-1}^2+a_{n+1}^2}{2}}$ for all $n \ge 1$ . Prove that this sequence is constant.

Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ satisfying: $x^2 f(x)+f(1-x)=2x-x^4$ for all $x \in \mathbb{R}$ .