HMMT-Feb 2017 guts, Problem 3

Find the number of pairs of integers $(x, y)$ such that $x^{2}+2 y^{2}<25$.

Solution: We do casework on $y$. If $y=0$, we have $x^{2}<25$, so we get 9 values of $x$. If $y= \pm 1$, then $x^{2}<23$, so we still have 9 values of $x$. If $y= \pm 2$, we have $x^{2}<17$, so we have 9 values of $x$. If $y= \pm 3$, we have $x^{2}<7$, we get 5 values of $x$. Therefore, the final answer is $9+2(9+9+5)=55$. $\fbox{55}$.

Answer: 55