SMO 1996

Three numbers are selected at random from the interval $[0,1]$. What is the probability that they form the lengths of the sides of a triangle?

Let $n$ be a positive integer. Prove that there is no positive integer solution to thxe equation $(x + 2)^n - x^n = 1 + 7^n$.

Determine all the solutions of the equation $x^3 + y^3 + z^3 = wx^2y^2z^2$ in natural numbers $x, y, z, w$. Justify your answer