Posted by: holdenlee | January 27, 2014

## Solving an inequality with entropy

The following is Problem 5 from IMO 1992:

Let $V$ be a finite subset of Euclidean space consisting of points $(x,y,z)$ with integer coordinates. Let $S_x,S_y,S_z$ be the projections of $V$ onto the $yz$, $xz$, $xy$ planes, respectively. Prove that

$|V|^2\le |S_x||S_y||S_z|.$

In this note, I’ll talk about how to solve this inequality using the idea of entropy. (source code)

## Responses

1. Cool notes!
Typo on page 4? 2 lg|V| = H(P), scratch the 2.

2. Thanks! I’ve fixed it.