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Divisible condition for every subset

Source: IV Caucasus Mathematic Olympiad

April 7, 2019
number theory

Problem Statement

Determine if there exist positive integers a1,a2,...,a10a_1,a_2,...,a_{10}, b1,b2,...,b10b_1,b_2,...,b_{10} satisfying the following property: for each non-empty subset SS of {1,2,…,10}\{1,2,\ldots,10\} the sum ∑i∈Sai\sum\limits_{i\in S}a_i divides (12+∑i∈Sbi)\left( 12+\sum\limits_{i\in S}b_i \right).
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