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 d_i, d_j, d_k.

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