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one of the numbers k and 100-k is representable

Source: 2017 Argentina OMA Finals L3 p2

January 16, 2023
Sumalgebranumber theoryconsecutive

Problem Statement

In a row there are 5151 written positive integers. Their sum is 100100 . An integer is representable if it can be expressed as the sum of several consecutive numbers in a row of 5151 integers. Show that for every kk , with 1≤k≤1001\le k \le 100 , one of the numbers kk and 100−k100-k is representable.
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