MathDB
A special polynomial condition

Source: Shortlist 2018 A6

July 17, 2019
algebrapolynomialIMO Shortlist

Problem Statement

Let m,n≥2m,n\geq 2 be integers. Let f(x1,…,xn)f(x_1,\dots, x_n) be a polynomial with real coefficients such that f(x1,…,xn)=⌊x1+⋯+xnm⌋ for every x1,…,xn∈{0,1,…,m−1}.f(x_1,\dots, x_n)=\left\lfloor \frac{x_1+\dots + x_n}{m} \right\rfloor\text{ for every } x_1,\dots, x_n\in \{0,1,\dots, m-1\}. Prove that the total degree of ff is at least nn.
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