MathNet 2009-part-01 — 175 problems (EN–VI) Question 30

A unit square is cut into $m$ quadrilaterals $Q_{1}, \ldots, Q_{m}$. For each $i=1, \ldots, m$ let $S_{i}$ be the sum of the squares of the four sides of $Q_{i}$. Prove that
$$
S_{1}+\ldots+S_{m} \geq 4
$$