MathDB

A6

Part of 2018 IMO Shortlist

Problems(1)

A special polynomial condition

Source: Shortlist 2018 A6

7/17/2019
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.
algebrapolynomialIMO Shortlist