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(another) Divisible condition for every subset

Source: IV Caucasus Mathematic Olympiad

April 7, 2019
number theory

Problem Statement

Determine if there exist pairwise distinct positive integers a1,a2,…,a101a_1,a_2,\ldots,a_{101}, b1b_1, b2b_2, \ldots, b101b_{101} satisfying the following property: for each non-empty subset SS of {1,2,…,101}\{1,2,\ldots,101\} the sum ∑i∈Sai\sum\limits_{i\in S}a_i divides (100!+∑i∈Sbi)\left( 100!+\sum\limits_{i\in S}b_i \right).
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