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Claim. Theorem 1 of Falikman's paper is van der Waerden's conjecture: every doubly stochastic $n\times n$ matrix $X$ satisfies $\operatorname{per}(X)\ge n!/n^n$. Theorem 2 adds that among the doubly stochastic matrices with all entries nonzero, the only one with $\operatorname{per}(A)=n!/n^n$ is the matrix whose entries are all $1/n$; the unrestricted equality case is not proved in this paper. The permanent is the sum over the $n!$ permutations $\sigma$ of the diagonal products $\prod_i x_{i\sigma(i)}$, so their average is at least $n^{-n}$ and some $\sigma$ attains $\prod_i x_{i\sigma(i)}\ge n^{-n}$, which is the statement of Problem 499. The argument minimizes the perturbed functional $\operatorname{per}(X)+\varepsilon/\prod_{i,j}x_{ij}$ over the doubly stochastic matrices with positive entries and lets $\varepsilon$ tend to zero; the library card summarizes it. Egorychev proved the same conjecture independently (claim page); the site also credits Gyires (claim page), whose paper does not prove the conjecture, so that attribution is rejected.

Acceptance. Refereed: Mat. Zametki 29 (1981), no. 6, 931–938, 957, with the English translation in Math. Notes 29 (1981), no. 6, 475–479; the issue is dated June 1981, and the page is dated to the first day of that month. The site's curator records the proofs of van der Waerden's conjecture but credits the problem's own statement to Marcus and Minc, whose earlier direct proof is the credited claim; this page therefore lists no reviewed evidence. The deduction from the permanent bound to the diagonal bound is the one-line averaging above.