Here's a Korean middle school midterm problem I've been struggling for quite some time now.
"$X$ is an irrational number such that $X>0$, and $Y$ is fractional part of $X$. If $$X^2+Y^2=27$$, find $X$."
So if we say
$X=(Y+N)$, such that $N=|X|$, we get
$(Y+N)^2+Y^2=27$; which is
As both $N^2$ and $27$ are rational, we must prove that there exists $Y$ and $N$ such that
$2(Y^2)+2NY$ is rational, which also means
$=XY$ is rational.
I can't find such irrational number which becomes a rational number when multiplied with its fractional part.
Can anyone tell how to solve this problem, or at least prove whether this is solvable/unsolvable?