Let a1,a2,…,an be positive reals for n≥2. For a permutation (b1,b2,…,bn) of (a1,a2,…,an), define its score to be n−1∑i=1b2ibi+1.Show that some two permutations of (a1,a2,…,an) have scores that differ by at most 3|a1−an|.
Source: Australia MO 2024 P3
Tags: algebra
Let a1,a2,…,an be positive reals for n≥2. For a permutation (b1,b2,…,bn) of (a1,a2,…,an), define its score to be n−1∑i=1b2ibi+1.Show that some two permutations of (a1,a2,…,an) have scores that differ by at most 3|a1−an|.