Generating functions of intersection numbers of certain tautological classes on moduli spaces of stable curves provide geometric solutions to integrable systems. Notable examples are the Kontsevich-Witten tau function and Brezin-Gross-Witten tau function. Both of them are tau-functions of the KdV hierarchy. Recently Mironov-Morozov gave a formula expressing Kontsevich-Witten tau function as a simple expansion of Schur's Q-polynomial with simple coefficients. This formula was called Mironov-Morozov conjecture by Alexandrov. A similar formula was also conjectured by Alexandrov for Brezin-Gross-Witten tau function. In this talk I will describe proofs of these formulas using Virasoro constraints. This is a joint work with Chenglang Yang.