MathDB

2

Part of 2022 Thailand Mathematical Olympiad

Problems(1)

Parity Inequality

Source: 2022 Thailand MO Day 1 P2

6/15/2022
Define a function f:N×N→{−1,1}f:\mathbb{N}\times \mathbb{N}\to\{-1,1\} such that f(m,n)={1if m,n have the same parity, and−1if m,n have different parityf(m,n)=\begin{cases} 1 &\text{if }m,n\text{ have the same parity, and} \\ -1 &\text{if }m,n\text{ have different parity}\end{cases} for every positive integers m,nm,n. Determine the minimum possible value of ∑1≤i<j≤2565ijf(xi,xj)\sum_{1\leq i<j\leq 2565} ijf(x_i,x_j) across all permutations x1,x2,x3,…,x2565x_1,x_2,x_3,\dots,x_{2565} of 1,2,…,25651,2,\dots,2565.
Inequalityalgebrainequalitiesnumber theory