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A Set of 100 Elements

Source: 1990 National High School Mathematics League, Exam Two, Problem 2

February 26, 2020

Problem Statement

E={1,2,⋯ ,200},G={a1,a2,⋯ ,a100}⊂EE=\{1,2,\cdots,200\},G=\{a_1,a_2,\cdots,a_{100}\}\subset E. GG satisfies the following: (1)For any 1≥i<j≥1001\geq i<j\geq100, a_i+a_j\neq201. (2)∑i=1100ai=10080\sum_{i=1}^{100}a_i=10080. Prove that the number of odd numbers in GG is a multiple of 44, and the sum the square of all numbers in GG is fixed.
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