1988 AMC 12 Problem 22

Attempt Problem 22 of the 1988 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 1988 AMC 12 solutions, or check the answer key.

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

22.

For how many integers

x

does a triangle with side lengths

10,

24,

and

x

have all its angles acute?

4

5

6

7

more than

7

Apply the strict Pythagorean inequality to the largest side in each possible ordering

The angle opposite

24

and the angle opposite

x

give the restrictive bounds

Acuteness requires

x^2<10^2+24^2=676

and

24^2<10^2+x^2,

x<26

and

x^2>476.

For integer

x,

this gives

22\le x\le25.

All four values also satisfy the triangle inequality, so there are

4

values.

Thus the correct answer is A.

Problem 22 in other years