MathDB

Derivatives, polynomials and functions

Source: 2021 Nigerian MO Round 3/P5

December 19, 2022

calculusderivativedegreePolynomialsalgebrapolynomialfunctions

Problem Statement

Let

f(x)=\frac{P(x)}{Q(x)}

, where

P(x), Q(x)

are two non-constant polynomials with no common zeros and

P(0)=P(1)=0

. Suppose

f(x)f\left(\frac{1}{x}\right)=f(x)+f\left(\frac{1}{x}\right)

for infinitely many values of

x

. a) Show that

\text{deg}(P)<\text{deg}(Q)

. b) Show that

P'(1)=2Q'(1)-\text{deg}(Q)\cdot Q(1)

.
Here,

P'(x)

denotes the derivative of

P(x)

as usual.