Putnam 2007 B5
Source:
December 3, 2007
Putnamalgebrapolynomialfloor functioncalculusderivativelinear algebra
Problem Statement
Let be a positive integer. Prove that there exist polynomials P_0(n),P_1(n),\dots,P_{k\minus{}1}(n) (which may depend on ) such that for any integer
\left\lfloor\frac{n}{k}\right\rfloor^k\equal{}P_0(n)\plus{}P_1(n)\left\lfloor\frac{n}{k}\right\rfloor\plus{} \cdots\plus{}P_{k\minus{}1}(n)\left\lfloor\frac{n}{k}\right\rfloor^{k\minus{}1}.
( means the largest integer )