Czechoslovak footballer Ivan Mráz began his playing career with Slovan Bratislava in 1959 before joining Spartak Praha in 1963. With spells at several other clubs Mráz eventually played more than 160 games in the Czechoslovak League, scoring 55 goals. After a brief spell with the Dutch team MVV Maastricht in 1969, he eventually finished his career with Dukla Prague in 1971. He made his international début for the Czechoslovakia football team in April 1964 and scored two goals in their 4–3 win against West Germany. Mráz had a short international career, playing only four matches, but scoring five goals, including two in his final match in November 1965. Czechoslovakia won silver at the 1964 Tokyo Olympics with Mráz scoring five goals in the tournament.
After his playing career Mráz turned to coaching, first in Czechoslovakia with Sparta Prague in 1974. He later moved to Costa Rica where he had two spells as the team coach of Liga Deportiva Alajuelense, winning two Costa Rican league titles in 1980 and 1991. Sandwiched between his stints with Alajuelense he also coached the Costa Rica national team in 1980.