MathDB

Putnam 1986 B3

Source:

August 5, 2019

Putnam

Problem Statement

Let

\Gamma

consist of all polynomials in

x

with integer coefficients. For

f

and

g

in

m

and

n

a positive integer, let

p

mean that every coefficient of

f-g

is an integral multiple of

m

. Let

n

and

p

be positive integers with

p

prime. Given that

f,g,h,r

and

s

are in

\Gamma

with

rf+sg\equiv 1 \pmod{p}

and

fg \equiv h \pmod{p}

, prove that there exist

F

and

G

in

\Gamma

with

F \equiv f \pmod{p}

,

G \equiv g \pmod{p}

, and

FG \equiv h \pmod{p^n}

.

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