MathDB
China Mathematics Olympiads (National Round) 2007 Problem 2

Source:

November 28, 2010
greatest common divisornumber theory unsolvednumber theory

Problem Statement

Show that: 1) If 2n−12n-1 is a prime number, then for any nn pairwise distinct positive integers a1,a2,…,ana_1, a_2, \ldots , a_n, there exists i,j∈{1,2,…,n}i, j \in \{1, 2, \ldots , n\} such that ai+aj(ai,aj)≥2n−1\frac{a_i+a_j}{(a_i,a_j)} \geq 2n-1 2) If 2n−12n-1 is a composite number, then there exists nn pairwise distinct positive integers a1,a2,…,ana_1, a_2, \ldots , a_n, such that for any i,j∈{1,2,…,n}i, j \in \{1, 2, \ldots , n\} we have ai+aj(ai,aj)<2n−1\frac{a_i+a_j}{(a_i,a_j)} < 2n-1
Here (x,y)(x,y) denotes the greatest common divisor of x,yx,y.
Back to ProblemsView on AoPS