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Algebra Difficulty 3.8 AMC 10/12 Find the answer United States

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

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

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 $9+2(9+9+5)=55$ .

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.