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Values of a polynomial

Source: Ibero-American Olympiad 1991, Problem 5

May 3, 2007

algebrapolynomialquadraticsnumber theory proposednumber theory

Problem Statement

Let

P(x,\, y)=2x^{2}-6xy+5y^{2}

. Let us say an integer number

a

is a value of

P

if there exist integer numbers

b

c

such that

P(b,\, c)=a

. a) Find all values of

P

lying between 1 and 100. b) Show that if

r

and

s

are values of

P

, then so is

rs