CMO 1983

Find all positive integers $w$ , $x$ , $y$ and $z$ which satisfy $w! = x! + y! + z!$ .

For each $r\in\mathbb{R}$ let $T_r$ be the transformation of the plane that takes the point $(x, y)$ into the point $(2^r x; r2^r x+2^r y)$ . Let $F$ be the family of all such transformations (i.e. $F = \{T_r : r\in\mathbb{R}\}$ ). Find all curves $y = f(x)$ whose graphs remain unchanged by every transformation in $F$ .

The area of a triangle is determined by the lengths of its sides. Is the volume of a tetrahedron determined by the areas of its faces?

Prove that for every prime number $p$ , there are infinitely many positive integers $n$ such that $p$ divides $2^n - n$ .

The geometric mean (G.M.) of $k$ positive integers $a_1$ , $a_2$ , $\dots$ , $a_k$ is defined to be the (positive) $k$ -th root of their product. For example, the G.M. of 3, 4, 18 is 6. Show that the G.M. of a set $S$ of $n$ positive numbers is equal to the G.M. of the G.M.'s of all non-empty subsets of $S$ .