2020 AIME I Problems/Problem 2
Note: Please do not post problems here until after the AIME.
Problem
There is a unique positive real number 
such that the three numbers 
, 
, and 
, in that order, form a geometric progression with positive common ratio. The number can be written as 
, where 
and 
are relatively prime positive integers. Find 
.
Solution
Since these form a geometric series, 
is the common ratio. Rewriting this, we get 
by base change formula. Therefore, the common ratio is 2. Now $\frac{\log_4{x}}{\log_8{2x}} = 2 \implies \log_4{x} = 2\log_8{2} + 2\log_8{x} \implies \frac{1}{2}\log_2{x} = \frac{2}{3} + \frac{2}{3}\log_2{x}
\implies -\frac{1}{6}\log_2{x} = \frac{2}{3} \implies \log_2{x} = -4 \implies x = \frac{1}{16}$ (Error compiling LaTeX. Unknown error_msg). Therefore, 
.
~ JHawk0224
See Also
| 2020 AIME I (Problems • Answer Key • Resources) | ||
| Preceded by Problem 1 |
Followed by Problem 3 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
| All AIME Problems and Solutions | ||
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