Menchik's greatest success at international tournaments was at Ramsgate 1929, when she finished tied for second with Akiba Rubinstein, just half a point behind former World Champion José Raúl Capablanca, and ahead of her teacher Géza Maróczy. In 1934 she finished third at Maribor, ahead of Rudolph Spielmann and Milan Vidmar. In 1942 she won a match against international master Jacques Mieses (four wins, five draws, one loss).