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)


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

  2. Thanks! I’ve fixed it.

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