In 1983 Szalma competed at the inaugural World Championships. He leapt 7.97 metres to progress from the qualifying round, and finished fourth in the final with 8.12 metres. He missed the 1984 Summer Olympics due to the Soviet-led 1984 Summer Olympics boycott, but repeated the fourth place at the 1985 World Indoor Games. During the 1984–85 indoor season he also won the silver medal at the European Indoor Championships. It was a very tight competition, with Szalma tying the winner, his compatriot Gyula Pálóczi, on 8.15 metres but with Pálóczi winning on countback. Also, Szalma was one centimetre ahead of bronze medalist Sergey Layevskiy and two centimetres ahead of fourth placer Ján Leitner. In the summer he jumped a career best of 8.30 metres, in July in Budapest. The result is the Hungarian record, and also the best result in Europe that year. At the end of the season he finished third at the 1985 World Cup, tying with but losing to Robert Emmiyan at 8.09 metres.