1995 AMC 12 Problem 23

Attempt Problem 23 of the 1995 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 1995 AMC 12 solutions, or check the answer key.

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

23.

The sides of a triangle have lengths

11,

15,

and

k,

where

k

is an integer. For how many values of

k

is the triangle obtuse?

5

7

12

13

14

First use the triangle inequality to bound the integer

k

Test the obtuse inequality separately when

15

is longest and when

k

is longest

The triangle inequality gives

5\le k\le25.

For

k\le15,

the longest side is

15,

and the triangle is obtuse when

15^2\gt11^2+k^2,

which gives

5\le k\le10,

6

values. For

k\gt15,

it is obtuse when

k^2\gt11^2+15^2=346,

giving

19\le k\le25,

7

values. The total is

6+7=13,

so the correct answer is D.

Problem 23 in other years