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sum a_i divides 100! + sum b_i

Source: 2021 Saudi Arabia Training Lists p39 https://artofproblemsolving.com/community/c2758131_2021_saudi_arabia_training_tests

January 5, 2022
Sumnumber theorydividesdivisible

Problem Statement

Determine if there exists pairwise distinct positive integers a1a_1, a2a_2,... ..., a101a_{101}, b1b_1, b2b_2,... ..., b101b_{101} satisfying the following property: for each non-empty subset SS of {1,2,...,101}\{1, 2, ..., 101\} the sum ∑i∈Sai\sum_{i \in S} a_i divides 100!+∑i∈Sbi100! + \sum_{i \in S} b_i.
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