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

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2015 AMC 10B Problem 12

Attempt Problem 12 of the 2015 AMC 10B below, then check your answer against the professionally curated solution 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 problems are used with official legal permission of the Mathematical Association of America (MAA).

12.

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

$11$

$12$

$13$

$14$

$15$

Answer: A

Concepts:coordinate geometry circle inequality

Difficulty rating: 1020

Solution:

The squared distance from $(x,-x)$ to $(5,5)$ is $\begin{aligned} &(x-5)^2+(-x-5)^2 \\ &= 2x^2+50. \end{aligned}$

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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Problem 12 in Other Years

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