Given any two numbers between $0$ and $1$ you have to prove that their sum is less than 1 plus their product; that is, given $0 < x < 1$ and $0 < y < 1$, prove that $x + y < 1 + xy$.

Hyeyoun Chung, St Paul's Girls' School, and Andaleeb Ahmed, Woodhouse Sixth Form College, London both produced nice solutions.

Consider $1-x$ and $1-y$. Since $0 < x < 1$ and $0 < y < 1$ it follows that

$\begin{eqnarray} \\ (1 - x)(1 - y) & > & 0 \\ 1 - x - y + xy & > & 0 \\ 1 + xy & > & x + y. \end{eqnarray}$

This is equivalent to $x + y < 1 + xy$.