MathDB

a^2+ab+ac-bc = 0, irrational, NT 2012 Romania District VII p2

Source:

September 1, 2024

algebranumber theoryirrational number

Problem Statement

Let

a, b

and

c

be positive real numbers such that

a^2+ab+ac-bc = 0.

a) Show that if two of the numbers

a, b

and

c

are equal, then at least one of the numbers

a, b

and

c

is irrational.
b) Show that there exist infinitely many triples

(m, n, p)

of positive integers such that

m^2 + mn + mp -np = 0.