2015 AMC 10B Problem 12 — Solution | LIVE by Po-Shen Loh

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2015 AMC 10B Problem 12

Below is the professionally curated solution for Problem 12 of the 2015 AMC 10B, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2015 AMC 10B solutions, or check the answer key.

All of the real AMC 8, AMC 10, AMC 12, and AIME problems in our complete solution collection are used with official legal permission of the Mathematical Association of America (MAA).

Concepts:coordinate geometry circle inequality

Difficulty rating: 1020

12.

For how many integers $x$ is the point $(x, -x)$ inside or on the circle of radius $10$ centered at $(5, 5)?$

$11$

$12$

$13$

$14$

$15$

Solution:

The squared distance from $(x,-x)$ to $(5,5)$ is $(x-5)^2+(-x-5)^2=2x^2+50.$

Being inside or on the circle means $2x^2+50\le100$ , so $x^2\le25$ . Thus $-5\le x\le5$ , giving $11$ integer values.

Thus, the correct answer is A.

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