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Pursuit racing is where two or more competitors (or teams) are either chasing after each other or chasing after a lead competitor or team.
The modern pentathlon is formatted such that the final event, a cross-country run, is essentially a pursuit race. Athlete performance in the prior four events determines the times in which they start the race. The idea is that the first athlete to cross the finish line wins the entire event.
In biathlon and cross-country skiing, this is the second part of a Sprint-Pursuit race where the Sprint winner will start first, whoever finished second by a certain time will follow the leader by that time. (For example, 1st finished 5 seconds ahead of 2nd in the Sprint event so 1st will start 5 seconds ahead of 2nd in the Pursuit event.)
In Nordic combined, the pursuit is determined by how well the competitors or team do in the ski jumping part of the competition. This is part of the Gundersen method developed in the 1980s. For the 15 km individual and 7.5 km sprint competitions, the computation is a 1 point advantage at the ski jump equals a 4 second advantage at the start of the cross-country skiing part of the competition. (For example, Competitor A finished with 5 more points than Competitor B in the ski jump, so Competitor B must start 20 seconds after Competitor A in the cross-country skiing part.) For the 4 x 5 km team event prior to 2005, the computation was a 1 point advantage at the ski jump equals a 1.5 second advantage in the cross-country skiing part of the competition. Since 2005, that computation is 1 point equals 1 second.
In cycling pursuit racing is a form of track racing, see:
Short- and long-track speed skating also have pursuit races.