The conjecture of van der Waerden, which states that the permanent of a doubly stochastic matrix is at least $n^{-n} n!$.

Let $M$ be a real $n \times n$ doubly stochastic matrix. Does there exist some $σ \in S_n$ such that $$\prod_{1 \leq i \leq n} M_{i, σ(i)} \geq n^{-n}?$$ This is true, and was proved by Marcus and Minc [MaMi62]

[MaMi62] Marcus, Marvin and Minc, Henryk, Some results on doubly stochastic matrices. Proc. Amer. Math. Soc. (1962), 571-579.