inequality similar to AM>GM

Question

Is the inequality $$\frac{1}{n^2}(x-y)^2<xy$$ true in $[0,1]$ for some fixed positive integer $n$ and $x\neq y$?

I couldn't seem to get the answer, though I have a hunch that it is not so!

Any kind of help is needed. Thanks!

Answer 1

It is false.

Let $x=0.5$ and $y=0.1$. Then $$(x-y)^2=0.4^2=0.16$$ but $$xy=0.5\times0.1=0.05$$

Answer 2

Take $x=3y$

$$(x-y)^2<xy \implies 4y^2<3y^2$$