Question

Let a and b be positive integers such that (1+ab)|(a2+b2).

Show that (a2+b2)/(1+ab) must be a perfect square.
I need some help in my IMO

Answer 1

To show that (a^2+b^2)/(1+ab) is a perfect square, we'll start by rewriting the expression and manipulating it. Since (1+ab) divides (a^2+b^2), it means (1+ab) divides ((a+b)^2). Therefore, (a^2+b^2)/(1+ab) is a perfect square.

Step-by-step explanation:

To show that (a^2+b^2)/(1+ab) is a perfect square, we'll start by rewriting the expression and manipulating it:

(a^2+b^2)/(1+ab) = (a^2+b^2+2ab-2ab)/(1+ab) = ((a^2+2ab+b^2)-2ab)/(1+ab)

= (a+b)^2/((1+ab)) - 2ab/(1+ab).

Since (1+ab) divides (a^2+b^2), it means (1+ab) divides ((a+b)^2).

Therefore, ((a^2+b^2)/(1+ab)) = ((a+b)^2/((1+ab)) - 2ab/(1+ab)) = ((a+b)^2)/((1+ab)) is a perfect square.