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China South East Mathematical Olympiad 2021 Grade11 P4

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August 8, 2021
number theoryPerfect Square

Problem Statement

For positive integer k,k, we say that it is a Taurus integer if we can delete one element from the set Mk={1,2,⋯ ,k},M_k=\{1,2,\cdots,k\}, such that the sum of remaining k−1k-1 elements is a positive perfect square. For example, 77 is a Taurus integer, because if we delete 33 from M7={1,2,3,4,5,6,7},M_7=\{1,2,3,4,5,6,7\}, the sum of remaining 66 elements is 25,25, which is a positive perfect square. (1)(1) Determine whether 20212021 is a Taurus integer. (2)(2) For positive integer n,n, determine the number of Taurus integers in {1,2,⋯ ,n}.\{1,2,\cdots,n\}.
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