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sum n_1/[n_i,n_{i+1}] <= 1- 1/2^{1999}, lcm inequality

Source: Singapore Senior Math Olympiad 2000 2nd Round p3 SMO

April 4, 2020
number theoryinequalitiesSumLCM

Problem Statement

Let n1,n2,n3,...,n2000n_1,n_2,n_3,...,n_{2000} be 20002000 positive integers satisfying n1<n2<n3<...<n2000n_1<n_2<n_3<...<n_{2000}. Prove that n1[n1,n2]+n1[n2,n3]+n1[n3,n4]+...+n1[n1999,n2000]≤1−121999\frac{n_1}{[n_1,n_2]}+\frac{n_1}{[n_2,n_3]}+\frac{n_1}{[n_3,n_4]}+...+\frac{n_1}{[n_{1999},n_{2000}]} \le 1 - \frac{1}{2^{1999}} where [a,b][a, b] denotes the least common multiple of aa and bb.
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