Difference between revisions of "2020 AIME I Problems/Problem 2"
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== Problem == | == Problem == | ||
| + | There is a unique positive real number <math>x</math> such that the three numbers <math>\log_8{2x}</math>, <math>\log_4{x}</math>, and <math>\log_2{x}</math>, in that order, form a geometric progression with positive common ratio. The number <math>x</math> can be written as <math>\frac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m + n</math>. | ||
== Solution == | == Solution == | ||
Revision as of 16:56, 12 March 2020
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 
. 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 | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
