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Show that 1/[a, b] + 1/[b, c] + 1/[c, d] + 1/[d, e] ≤ 15/16

Source: Canada National Mathematical Olympiad 1979 - Problem 3

September 29, 2011
least common multiplenumber theory proposednumber theory

Problem Statement

Let aa, bb, cc, dd, ee be integers such that 1≤a<b<c<d<e1 \le a < b < c < d < e. Prove that 1[a,b]+1[b,c]+1[c,d]+1[d,e]≤1516,\frac{1}{[a,b]} + \frac{1}{[b,c]} + \frac{1}{[c,d]} + \frac{1}{[d,e]} \le \frac{15}{16}, where [m,n][m,n] denotes the least common multiple of mm and nn (e.g. [4,6]=12[4,6] = 12).
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