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.

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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

Substitute

(x,-x)

into the distance formula from

(5,5)

Solve

2x^2+50\le100

The squared distance from

(x,-x)

(5,5)

\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.

Problem 12 in other years