5

Part of 2007 Putnam

Problems(2)

Source:

Suppose that a finite group has exactly

n

elements of order

p,

p

is a prime. Prove that either n\equal{}0 or

p

divides n\plus{}1.

Putnamgroup theoryabstract algebrainductiongeometrygeometric transformationrotation

Source:

k

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

k

) such that for any integer

n,

\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}. (

\lfloor a\rfloor

means the largest integer

\le a.

Putnamalgebrapolynomialfloor functioncalculusderivativelinear algebra