Difference between revisions of "1969 AHSME Problems/Problem 25"
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== Solution == | == Solution == | ||
| − | <math>\ | + | We use the logarithm property of addition: |
| + | <cmath> | ||
| + | \begin{align*} | ||
| + | \log_2(a)+\log_2(b) \ge 6 &= \log_2(ab) \ge 6\\ | ||
| + | &\Rightarrow 2^{log_2(ab)} \ge 2^6\\ | ||
| + | &= ab \ge 64 | ||
| + | \end{align*}</cmath> | ||
| + | Due to the Quadratic [[Optimization]] or the [[AM-GM Inequality]], the least value is obtained when <math>a = b</math>. | ||
| + | Therefore, <math>a = b = 8 \Rightarrow a + b = \boxed{(D)16}</math> | ||
== See also == | == See also == | ||
Latest revision as of 16:59, 21 April 2020
Problem
If it is known that 
, then the least value that can be taken on by 
is:
Solution
We use the logarithm property of addition:
Due to the Quadratic Optimization or the AM-GM Inequality, the least value is obtained when 
.
Therefore, 
See also
| 1969 AHSC (Problems • Answer Key • Resources) | ||
| Preceded by Problem 24 |
Followed by Problem 26 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 • 31 • 32 • 33 • 34 • 35 | ||
| All AHSME Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
