MathDB
{i, j, k} = {1, 2, 3}, (x \in A_i, y\in A_j) > (x + y \in A_k, x - y \in A_k).

Source: Czech and Slovak Olympiad 1985, National Round, Problem 2

September 11, 2024
Subsetscombinatoricsnumber theory

Problem Statement

Let A1,A2,A3A_1, A_2, A_3 be nonempty sets of integers such that for {i,j,k}={1,2,3}\{i, j, k\} = \{1, 2, 3\} holds (x∈Ai,y∈Aj)⇒(x+y∈Ak,x−y∈Ak).(x \in A_i, y\in A_j) \Rightarrow (x + y \in A_k, x - y \in A_k). Prove that at least two of the sets A1,A2,A3A_1, A_2, A_3 are equal. Can any of these sets be disjoint?
Back to ProblemsView on AoPS