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Numbers on board [All-Russian 2017, Day 1, G11, P3]

Source: All Russian Olympiads 2017, Day1, Grade 11, Problem 3

May 1, 2017
number theoryalgebra

Problem Statement

There are nn positive real numbers on the board a1,…,ana_1,\ldots, a_n. Someone wants to write nn real numbers b1,…,bnb_1,\ldots,b_n,such that: bi≥aib_i\geq a_i If bi≥bjb_i \geq b_j then bibj\frac{b_i}{b_j} is integer. Prove that it is possible to write such numbers with the condition b1⋯bn≤2n−12a1⋯an.b_1 \cdots b_n \leq 2^{\frac{n-1}{2}}a_1\cdots a_n.
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