2007
Problem - 2343
Let $ n$ be a positive integer and $ [ \ n ] = a.$ Find the largest integer $ n$ such that the following two conditions are satisfied:
$ (1)$ $ n$ is not a perfect square;
$ (2)$ $ a^{3}$ divides $ n^{2}$.
Let $ n = p^2 + r$ where $ 0\leq r < 2p + 1$ $ \Longrightarrow [\sqrt {n}] = p$ Imply that : $ p^3|(p^2 + r)^2$ (1) So $ p|r,\Longrightarrow r\in \{0,p,2p\}$
Case 1 $ r = 0$ Easy to check it is true for all p . So $ n = p^2$
Case 2 $ r = p$ (1)$ \Longleftrightarrow p^3|(p^2 + p)^2$ $ \Longleftrightarrow p|(p + 1)^2$ $ \Longleftrightarrow p = 1$ So $ n\in \{1,2\}$
Case 3 $ r = 2p$ (1)$ \Longleftrightarrow p|(p + 2)^2$ So $ p|4$ So $ n\in \{3,8,15,24\}$
So solution is $ n\in \{t^2,1,2,3,8,15,24\}$