2019 AMC 12B Problem 9 — Solution | LIVE by Po-Shen Loh

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2019 AMC 12B Problem 9

Below is the professionally curated solution for Problem 9 of the 2019 AMC 12B, 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 of the real AMC 8, AMC 10, AMC 12, and AIME problems in our complete solution collection are used with official legal permission of the Mathematical Association of America (MAA).

Concepts:logarithm triangle inequality counting integers in a range

Difficulty rating: 1500

9.

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 $3?$

$57$

$59$

$61$

$62$

$63$

Solution:

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.

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