1965 AMC 12 Problem 13

Attempt Problem 13 of the 1965 AMC 12 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 1965 AMC 12 solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

13.

Let

n

be the number of number-pairs

(x,y)

which satisfy

5y-3x=15

and

x^2+y^2\leq16.

Then

n

is:

0

1

2

more than two, but finite

greater than any finite number

The inequality describes the disk of radius

4

centered at the origin

Compare the line’s distance

\frac{15}{\sqrt{34}}

from the origin with

4

The line’s distance from the origin is

\frac{15}{\sqrt{(-3)^2+5^2}}=\frac{15}{\sqrt{34}}\lt4.

It therefore crosses the interior of the disk

x^2+y^2\leq16

in a segment containing infinitely many points.

Thus, the correct answer is E.

Problem 13 in other years