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A8. Acute triangles with given side lengths

This is a neat problem from last year's Putnam Examination. It was published (with answer given) in the most recent MAA Monthly.

Given 12 real numbers d1, ... , d12 on the interval (1, 12), show that there exist distinct indices i, j, k such that there is an acute triangle with side lengths di, dj, dk.

I will post a hint as a comment in a few days.