MathDB
\sum x_k -\sum |x_k| is divisible by 4 , if (-1)^{x_k} x_{k-1}x_{k+1} >0

Source: 1982 Polish MO Finals p5

February 25, 2020
divisibleSequenceSum

Problem Statement

Integers x0,x1,...,xn−1,xn=x0,xn+1=x1x_0,x_1,...,x_{n-1}, x_n = x_0, x_{n+1} = x_1 satisfy the inequality (−1)xkxk−1xk+1>0(-1)^{x_k} x_{k-1}x_{k+1} >0 for k=1,2,...,nk = 1,2,...,n. Prove that the difference ∑k=0n−1xk−∑k=0n−1∣xk∣\sum_{k=0}^{n-1}x_k -\sum_{k=0}^{n-1}|x_k| is divisible by 44.
Back to ProblemsView on AoPS