IL-GILLIS 1995

Solve the system $$\begin{cases} x+\log\left(x+\sqrt{x^2+1}\right)=y y+\log\left(y+\sqrt{y^2+1}\right)=z z+\log\left(z+\sqrt{z^2+1}\right)=x \end{cases}$$

Let $PQ$ be the diameter of semicircle $H$ . Circle $O$ is internally tangent to $H$ and tangent to $PQ$ at $C$ . Let $A$ be a point on $H$ and $B$ a point on $PQ$ such that $AB\perp PQ$ and is tangent to $O$ . Prove that $AC$ bisects $\angle PAB$

If $k$ and $n$ are positive integers, prove the inequality $$\frac{1}{kn} +\frac{1}{kn+1} +...+\frac{1}{(k+1)n-1} \ge n \left(\sqrt[n]{\frac{k+1}{k}}-1\right)$$

Find all integers $m$ and $n$ satisfying $m^3 -n^3 - 9mn = 27$ .

Let $n$ be an odd positive integer and let $x_1,x_2,...,x_n$ be n distinct real numbers that satisfy $|x_i -x_j| \le 1$ for $1 \le i < j \le n$ . Prove that $$\sum_{i<j} |x_i -x_j| \le \left[\frac{n}{2} \right] \left(\left[\frac{n}{2} \right]-1 \right)$$

A $1995 \times 1995$ square board is given. A coloring of the cells of the board is called good if the cells can be rearranged so as to produce a colored square board that is symmetric with respect to the main diagonal. Find all values of $k$ for which any $k$ -coloring of the given board is good.

For certain $n$ countries there is an airline connecting any two countries, but some of the airlines are closed. Show that if the number of the closed airlines does not exceed $n-3$ , then one can make a round trip using the remaining airlines, starting from one of the countries, visiting every country exactly once and returning to the starting country.

A real number $\alpha$ is given. Find all functions $f : R^+ \to R^+$ satisfying $\alpha x^2f\left(\frac{1}{x}\right) +f(x) =\frac{x}{x+1}$ for all $x > 0$ .