problem:Find all function G from the set of natural numbers to natural numbers such that (g(m) +n)(g(n)+m) is a perfect square for all m,n (natural numbers) .
solution: Lets take g(m) = m+ f(m)
The given expression equals (m+n)2+ (m+n)(f(m) + f(n)) + f(m)f(n)
We see this as a quadratic on (m+n). Since the expression gives perfect square for infinitely many values, we must have its discriminant equal to zero.
That gives f(m)= f(n) hence f is constant
we obtain solutions G(t) = t+ C, obviously C has to be positive
Find flaw in above solution.(No actual solution needed, discuss the flaw)
341I can understand the conclusion that if an^2+bn+c is a perfect square for all n, then it has to be of the form (pn+q)2
But with f(m) and f(n) also variables how is this conclusion valid?
Just to check - you are aware that this is IMO 2010, right?
62We see this as a quadratic on (m+n). Since the expression gives perfect square for infinitely many values, we must have its discriminant equal to zero.
I dont think so!
21Yep
I was aware that this is IMO 2010
I forgot that coefficients are not constant
I was sure of the mistake but could not trace it that time..
Thank u prophet sir, Nishant sir