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\frac{P(x+1)-P(x)}{P(x+\pi)}= \frac{a}{x+\pi}, prove a is a natural

Source: INAMO Shortlist 2015 A2

May 4, 2019

polynomialfunctional equationNatural Numberalgebra

Problem Statement

Suppose

a

real number so that there is a non-constant polynomial

P (x)

such that

\frac{P(x+1)-P(x)}{P(x+\pi)}= \frac{a}{x+\pi}

for each real number

x

, with

x+\pi \ne 0

and

P(x+\pi)\ne 0

. Show that

a

is a natural number.