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grasshopper jumps on the real line from 0 to 2010

Source: OIFMAT I 2010 day 2 p4 - Chilean Math Forum FMAT Olympiad https://artofproblemsolving.com/community/c2484778_oifmat

September 25, 2021
number theory

Problem Statement

Let a1<a2<...<an a_1 <a_2 <... <a_n consecutive positive integers (with n>2 n> 2 ). A grasshopper jumps on the real line, starting at point 0 0 and jumping n n to the right with lengths a1 a_1 , a2 a_2 , ..., an a_n , in some order (each length occupies exactly once), ending your tour at the 2010 2010 point. Find all the possible values n n of jumps that the grasshopper could have made.
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