Find all continuous positive functions $f$, if any, defined on $[0,1]$ so that:
$\int_0^1$$f(x)dx=1,$ $\int_0^1$$f(x)$$\sin$$\pi$$x$dx $=$$a$$,$ $\int_0^1$$f(x)$$\sin^2$$\pi$$x$dx $=$$a^2$
where $a$ is a given whole number.
Attempt
If we use integration by part and take $u$ as $\sin^2$$\pi$$x$ and $dv$ as $f(x)dx$, then we will get $a=0$ if we solve the third integral.
The problem has been with the second integral. Choosing $u$ as $\sin$$\pi$$x$ leads nowhere and choosing $u$ as $f(x)$ does not help matters as well.
I will really appreciate it if you guys can suggest a way out for me.
Thanks.