4

Part of 2007 Putnam

Problems(2)

Source:

A repunit is a positive integer whose digits in base

10

are all ones. Find all polynomials

f

with real coefficients such that if

n

is a repunit, then so is

f(n).

Putnamalgebrapolynomiallogarithmsquadraticscollege contests

Source:

n

be a positive integer. Find the number of pairs

P,Q

of polynomials with real coefficients such that (P(X))^2\plus{}(Q(X))^2\equal{}X^{2n}\plus{}1 and

\text{deg}P<\text{deg}{Q}.

Putnamalgebrapolynomialcollege contests