MathDB

8

Part of 2019 Caucasus Mathematical Olympiad

Problems(2)

Divisible condition for every subset

Source: IV Caucasus Mathematic Olympiad

4/7/2019
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).
number theory
(another) Divisible condition for every subset

Source: IV Caucasus Mathematic Olympiad

4/7/2019
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).
number theory