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Natalia and Marcela counting numbers

Source: II May Olympiad (Olimpiada de Mayo) 1996 L2 P3

September 17, 2022
algebranumber theory

Problem Statement

Natalia and Marcela count 11 by 11 starting together at 11, but Marcela's speed is triple that of Natalia (when Natalia says her second number, Marcela says the fourth number). When the difference of the numbers that they say in unison is any of the multiples of 29 29, between 500500 and 600600, Natalia continues counting normally and Marcela begins to count downwards in such a way that, at one point, the two say in unison the same number. What is said number?