Whilst very few people doubt that there are exciting new discoveries awaiting the inspired researcher in Physics, or Chemistry, or almost any other science, many seem to imagine that virtually all mathematics has been known for fifty years or more.
J11 A drill squad contains $mn$ people, arranged in $m$ rows of $n$ people. In each of the $n$ columns so formed, the lightest person is noted, and the heaviest of these is found to be Smith.
J1 (a) Two numbers, $a$ and $b$ are such that $a$ is smaller and $b$ is greater than $1$. If $S$ is the sum of $a$ and $b$, and $P$ is their product, prove that $S$ and $P$ differ by more than $1$.