IL-GILLIS 1997

Find all real solutions to the system of equations $$\begin{cases} x^2 +y^2 = 6z y^2 +z^2 = 6x z^2 +x^2 = 6y \end{cases}$$

We are given a balance with two bowls and a number of weights. (a) Give an example of four integer weights using which one can measure any weight of $1,2,...,40$ grams. (b) Are there four weights using which one can measure any weight of $1,2,...,50$ grams?

Let $n?$ denote the product of all primes smaller than $n$ . Prove that $n? > n$ holds for any natural number $n > 3$ .

Let $f : [0,1] \to [0,1]$ be a continuous, strictly increasing function such that $f(0) = 0$ and $f(1) = 1$ . Prove that $$f\left(\frac{1}{10}\right) + f\left(\frac{2}{10}\right) +...+f\left(\frac{9}{10}\right) +f^{-1}\left(\frac{1}{10}\right) +...+f^{-1}\left(\frac{9}{10}\right) \le \frac{99}{10}$$

The natural numbers $a_1,a_2,...,a_n, n \ge 12$ , are smaller than $9n^2$ and pairwise coprime. Show that at least one of these numbers is prime.

In a certain country, every two cities are connected either by an airline route or by a railroad. Prove that one can always choose a type of transportation in such a way that each city can be reached from any other city with at most two transfers.

A square with side $10^6$ , with a corner square with side $10^{-3}$ cut off, is partitioned into $10$ rectangles. Prove that at least one of these rectangles has the ratio of the greater side to the smaller one at least $9$ .

Two equal circles are internally tangent to a larger circle at $A$ and $B$ . Let $M$ be a point on the larger circle, and let lines $MA$ and $MB$ intersect the corresponding smaller circles at $A'$ and $B'$ . Prove that $A'B'$ is parallel to $AB$ .