MathDB
Non-trivial sequence, do you exist?

Source: BdMO 2024 Secondary National P4 Higher Secondary National P4

March 18, 2024
combinatoricsSequencemodular arithmeticpigeonhole principle

Problem Statement

Let a1,a2,…,a11a_1, a_2, \ldots, a_{11} be integers. Prove that there exist numbers b1,b2,…,b11b_1, b_2, \ldots, b_{11} such that
[*] bib_i is equal to −1,0-1,0 or 11 for all i∈{1,2,…,11}i \in \{1, 2,\dots, 11\}. [*] all numbers can't be zero at a time. [*] the number N=a1b1+a2b2+…+a11b11N=a_1b_1+a_2b_2+\ldots+a_{11}b_{11} is divisible by 20242024.
Back to ProblemsView on AoPS