MathDB
JBMO Shortlist 2022 N2

Source: JBMO Shortlist 2022

June 26, 2023
number theoryLCMLowest common multipleJuniorBalkanshortlistInequality

Problem Statement

Let a<b<c<d<ea < b < c < d < e be positive integers. Prove that 1[a,b]+1[b,c]+1[c,d]+2[d,e]≤1\frac{1}{[a, b]} + \frac{1}{[b, c]} + \frac{1}{[c, d]} + \frac{2}{[d, e]} \le 1 where [x,y][x, y] is the least common multiple of xx and yy (e.g., [6,10]=30[6, 10] = 30). When does equality hold?
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