OME 2003

Prove that for any prime ${p}$ , different than ${2}$ and ${5}$ , there exists such a multiple of ${p}$ whose digits are all nines. For example, if ${p = 13}$ , such a multiple is ${999999 = 13 * 76923}$ .

Does there exist such a finite set of real numbers ${M}$ that has at least two distinct elements and has the property that for two numbers, ${a}$ , ${b}$ , belonging to ${M}$ , the number ${2a - b^2}$ is also an element in ${M}$ ?

The altitudes of the triangle ${ABC}$ meet in the point ${H}$ . You know that ${AB = CH}$ . Determine the value of the angle $\widehat{BCA}$ .

Let ${x}$ be a real number such that ${x^3 + 2x^2 + 10x = 20.}$ Demonstrate that both ${x}$ and ${x^2}$ are irrational.

How many possible areas are there in a convex hexagon with all of its angles being equal and its sides having lengths $1, 2, 3, 4, 5$ and $6,$ in any order?

We string $2n$ white balls and $2n$ black balls, forming a continuous chain. Demonstrate that, in whatever order the balls are placed, it is always possible to cut a segment of the chain to contain exactly $n$ white balls and $n$ black balls.