AMC 12 Mock Exam 1 Problem 17

Write $x=n+f$ with $n=\lfloor x\rfloor\in\{1,\dots,9\}$ and $f=\{x\}\in[0,1)$. Then $x^2=n^2+2nf+f^2$.

Claim: $\{x^2\}=f^2$ if and only if $2nf$ is an integer. If $2nf=m\in\mathbb Z$, then $x^2=(n^2+m)+f^2$ with $0\le f^2<1$, so $\{x^2\}=f^2$. Conversely, if $\{x^2\}=f^2$, then $x^2-f^2=n^2+2nf$ equals $\lfloor x^2\rfloor$, an integer, so $2nf\in\mathbb Z$.

For fixed $n$, the values $f=\frac{k}{2n}$ with $k=0,1,\dots,2n-1$ are exactly the $2n$ numbers in $[0,1)$ with $2nf\in\mathbb Z$. Hence the count is $\sum_{n=1}^{9}2n=2\cdot45=\boxed{90}$.