In 1965, he tied for 7–8th in Moscow (Central Chess Club-ch; Vladimir Liberzon won).
In 1967, he tied for 41–57th in Kharkov (35th USSR Championship; Lev Polugaevsky and Mikhail Tal won).
In 1978, he was one of Viktor Korchnoi's seconds in Baguio during a match for the World Chess Championship against Anatoly Karpov.
Murey played for Israel in three Chess Olympiads:[2] In 1982/83, he tied for 3rd–4th in Hastings (Rafael Vaganian won).
He also played for Israel in 1st World Senior Team Championship at Isle of Man 2004.
He won gold team medal[3] and was equal 1st with Wolfgang Uhlmann on first board there.
In a 1993 game against Jan Timman, Murey debuted a surprising novelty in the Petrov's Defense, a temporary knight sacrifice with 4.
Though Murey lost the game, the discovery is considered sound, and is frequently referred to in opening databases as the Murrey [sic] Variation of the Modern Attack in the Petrov's defense.