IMO 2019 Shortlist A3
Let n ě 3 be a positive integer and let pa1,a2,...,anq be a strictly increasing sequence of n positive real numbers with...
Category: Algebra
Problem
Let n ě 3 be a positive integer and let pa1,a2,...,anq be a strictly increasing sequence of n positive real numbers with sum equal to 2. Let X be a subset of t1,2,...,nu such that the value of ˇ ˇ ˇ ˇ ˇ 1 ´ ÿ iPX ai ˇ ˇ ˇ ˇ ˇ is minimised. Prove that there exists a strictly increasing sequence of n positive real numbers pb1,b2,...,bnq with sum equal to 2 such that ÿ iPX bi “ 1. (New Zealand)