Problem 6

Determine the least possible value of

f(1998)

, where

f:\mathbb{N}\to\mathbb{N}

is a function such that for all

m,n\in\mathbb{N}

f\left(n^{2}f(m)\right)=m\left(f(n)\right)^{2}.

Step 1 of 6: Extract the first identities

In plain words

Special substitutions reveal the scaling constant.

f(kt^2)=f(t)^2,\quad f(f(t))=k^2t,\quad k=f(1)

Detailed analysis

Setting $m=1$ gives $f(kt^2)=f(t)^2$ , while setting $n=1$ gives $f(f(t))=k^2t$ . Applying these identities to $t$ and $kt$ yields $f(kt)=kf(t)$ .