He was born in Nagymaros. He won a Hungarian title for the first time in 1977, taking the national indoor championship, and first made his mark in international athletics at the 1977 European Indoor Championships. With a jump of 7.78 metres he won the bronze medal. At the 1978 European Indoor Championships the next year he won the gold medal with a jump of 7.83 metres. In 1980 he competed at the Olympic Games in Moscow, finishing fourth. Then, despite jumping even better at the 1981 European Indoor Championships, Szalma only finished fourth with 7.90 metres, eleven centimetres behind the winner Rolf Bernhard. In the summer he won the gold medal at the 1981 Summer Universiade. At the 1982 European Indoor Championships, there was an eleven centimetre gap between first and seventh place; Szalma finished in between at a fifth place. At the 1982 European Championships he dropped slightly to an eleventh place.