What is the smallest integer greater than 1 such that $\frac12$ of it is a perfect square and $\frac15$ of it is a perfect fifth power? 
What is the smallest integer greater than 1 such that $\frac12$ of it is a perfect square and $\frac15$ of it is a perfect fifth power?

I have tried multiplying every perfect square (up to 400 by two and checking if it is a perfect 5th power, but still nothing. I don't know what to do at this point.

All integers of this kind can be written in the form,
$$k = 5^{10a - 4} 2^{10b - 5} c^{10d}$$
where $a, b, c, d \in \mathbb{Z}_{\ge 1}$ (or $a, b, c, d$ are non-negative integers)

Let's make sure this works.
$$ k/5 = 5^{10a-5} 2^{10b-5} = (5^{2a-1} 2^{2b-1} c^{2d})^5$$
So, $1/5$ of $k$ is a perfect fifth power.
$$ k/2 = 5^{10a-4} 2^{10b-6} = (5^{5a-2} 2^{5b-3} c^{5d})^2$$
So, $1/2$ of $k$ is a perfect square.

The smallest integer of this kind is the one where $a, b, c, d = 1$ which is $k = 5^6 2^5 = 500000$.

You can find all integers that follow your definition by using different values of $a, b, c, d$. 
