Maths Problems Thread

[Roflcopter]

Spoiler

2^4 & 4^2. Only one I can answer off the top of my head.

Nice solution for the non-trivial solutions.

Spoiler

Since the question didn't ask for proof I think it's solved. Here is the proof I came to:

We note immediately that x=y is a solution for all positive integers.

We'll now further restrict the question to x<y. Then and since both x and y are integers y must divide x, that is, . So we get which implies k=2, which in turn decides x=2 and y=4.

Hence all solutions are (2, 4), (4, 2) and (λ, λ), λ a positive integer.

Spoiler

I didn't prove that implies k=2 above but it's fairly intuitive. First prove that the function is monotonically decreasing for k>1 (high-school calculus will do). Then note that for k>2 we have 1<f(k)<2 so f(k) can never be equal to an integer.