OBM 1989

The sides of a triangle $T$ , with vertices $(0,0)$ , $(3,0)$ and $(0,3)$ are mirrors.

Show that one of the images of the triagle $T_1$ with vertices $(0,0)$ , $(0,1)$ and $(2,0)$ is the triangle with vertices $(24,36)$ , $(24,37)$ and $(26,36)$ .

Let $k$ a positive integer number such that $\frac{k(k+1)}{3}$ is a perfect square. Show that $\frac{k}{3}$ and $k+1$ are both perfect squares.

A function $f$ , defined for the set of integers, is such that $f(x)=x-10$ if $x>100$ and $f(x)=f(f(x+11))$ if $x \leq 100$ .

Determine, justifying your answer, the set of all possible values for $f$ .

A game is played by two contestants A and B, each one having ten chips numbered from 1 to 10. The board of game consists of two numbered rows, from 1 to 1492 on the first row and from 1 to 1989 on the second.

At the $n$ -th turn, $n=1,2,\ldots,10$ , A puts his chip numbered $n$ in any empty cell, and B puts his chip numbered $n$ in any empty cell on the row not containing the chip numbered $n$ from A.

B wins the game if, after the 10th turn, both rows show the numbers of the chips in the same relative order. Otherwise, A wins.

[list=a] [] Which player has a winning strategy? [] Suppose now both players has $k$ chips numbered 1 to $k$ . Which player has a winning strategy? [*] Suppose further the rows are the set $\mathbb{Q}$ of rationals and the set $\mathbb{Z}$ of integers. Which player has a winning strategy? [/list]

A tetrahedron is such that the center of the its circumscribed sphere is inside the tetrahedron.

Show that at least one of its edges has a size larger than or equal to the size of the edge of a regular tetrahedron inscribed in this same sphere.