2019 AMC 12B Problem 9

Attempt Problem 9 of the 2019 AMC 12B 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 2019 AMC 12B solutions, or check the answer key.

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

For how many integral values of

x

can a triangle of positive area be formed having side lengths

\log_2 x,

\log_4 x,

and

3?

57

59

61

62

63

Let

t=\log_2 x,

\log_4 x=\dfrac{t}{2}

Apply the triangle inequality to sides

t,\ \dfrac{t}{2},\ 3

Let

t=\log_2 x.

Then

\log_4 x=\dfrac{t}{2},

and the sides are

t,\ \dfrac{t}{2},\ 3.

The triangle inequalities give

t+\dfrac{t}{2}\gt3

(so

t\gt2

) and

\dfrac{t}{2}+3\gt t

(so

t\lt6

); the third inequality is automatic.

Thus

2\lt\log_2 x\lt6,

i.e.

4\lt x\lt64.

The integers

5,6,\ldots,63

number

59.

Thus, B is the correct answer.

Problem 9 in other years