IMO 1968

Find all triangles whose side lengths are consecutive integers, and one of whose angles is twice another.

Find all natural numbers $n$ the product of whose decimal digits is $n^2-10n-22$ .

Prove that every tetrahedron has a vertex whose three edges have the right lengths to form a triangle.

Let $f$ be a real-valued function defined for all real numbers, such that for some $a>0$ we have $$f(x+a)={1\over2}+\sqrt{f(x)-f(x)^2}$$ for all $x$ . Prove that $f$ is periodic, and give an example of such a non-constant $f$ for $a=1$ .