MathDB
Existence of alpha and beta satisfying min condition

Source: Serbia 2024 MO Problem 3

April 4, 2024
algebra

Problem Statement

Let nn be a positive integer and let a1,a2,…,ana_1, a_2, \ldots, a_n and b1,b2,…,bnb_1, b_2, \ldots, b_n be reals. Show that for any positive integer 1≤m≤n1 \leq m \leq n, there exist two distinct reals α,β\alpha, \beta, α2+β2>0\alpha^2+\beta^2>0, such that pm=min⁡{p1,p2,…,pn}p_m=\min\{p_1, p_2, \ldots, p_n\}, where pj=∑i=1n∣α(ai−aj)+β(bi−bj)∣p_j=\sum_{i=1}^n|\alpha(a_i-a_j)+\beta(b_i-b_j)| for 1≤j≤n1\leq j \leq n.
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