Difference between revisions of "2012 AIME I Problems/Problem 9"
(Created page with "== Problem 9 == Let <math>x,</math> <math>y,</math> and <math>z</math> be positive real numbers that satisfy <cmath>2\log_{x}(2y) = 2\log_{2x}(4z) = \log_{2x^4}(8yz) \ne 0.</cmat...") |
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== Solution == | == Solution == | ||
+ | Since there are only two dependent equations given and three unknowns, the three expressions given can equate to any common value, so to make the problem as simple as possible let us assume with loss of generality that | ||
+ | <cmath>2\log_{x}(2y) = 2\log_{2x}(4z) = \log_{2x^4}(8yz) = 2.</cmath> | ||
+ | Then | ||
+ | <cmath> | ||
+ | \begin{align*} | ||
+ | 2\log_{x}(2y) = 2 &\longrightarrow x=2y\ | ||
+ | 2\log_{2x}(4z) = 2 &\longrightarrow 2x=4z\ | ||
+ | \log_{2x^4}(8yz) = 2 &\longrightarrow 4x^8 = 8yz | ||
+ | \end{align*} | ||
+ | </cmath> | ||
+ | Solving these equations, we quickly see that <math>4x^8 = (2y)(4z) = x(2x) \longrightarrow x=2^{-1/6}</math> and then <math>y=z=2^{-1/6 - 1} = 2^{-7/6}.</math> | ||
+ | Finally, our desired value is <math>2^{-1/6} \cdot (2^{-7/6})^5 \cdot 2^{-7/6} = 2^{-43/6}</math> and thus <math>m+n = 43 + 6 = \boxed{049.}</math> | ||
== See also == | == See also == | ||
{{AIME box|year=2012|n=I|num-b=8|num-a=10}} | {{AIME box|year=2012|n=I|num-b=8|num-a=10}} |
Revision as of 01:22, 17 March 2012
Problem 9
Let and be positive real numbers that satisfy The value of can be expressed in the form where and are relatively prime positive integers. Find
Solution
Since there are only two dependent equations given and three unknowns, the three expressions given can equate to any common value, so to make the problem as simple as possible let us assume with loss of generality that Then Solving these equations, we quickly see that and then Finally, our desired value is and thus
See also
2012 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 8 |
Followed by Problem 10 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |