Difference between revisions of "2010 AIME I Problems/Problem 3"
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== Problem == | == Problem == | ||
| + | Suppose that <math>y = \frac34x</math> and <math>x^y = y^x</math>. The quantity <math>x + y</math> can be expressed as a rational number <math>\frac {r}{s}</math>, where <math>r</math> and <math>s</math> are relatively prime positive integers. Find <math>r + s</math>. | ||
| + | |||
| + | == Solution == | ||
| + | We solve in general using <math>c</math> instead of <math>3/4</math>. Substituting <math>y = cx</math>, we have: | ||
| + | |||
| + | <center><cmath>x^{cx} = (cx)^x \Longrightarrow (x^x)^c = c^x\cdot x^x</cmath></center> | ||
| + | |||
| + | Dividing by <math>x^x</math>, we get <math>(x^x)^{c - 1} = c^x</math>. | ||
| + | |||
| + | Taking the <math>x</math>th root, <math>x^{c - 1} = c</math>, or <math>x = c^{1/(c - 1)}</math>. | ||
| + | |||
| + | In the case <math>c = \frac34</math>, <math>x = \Bigg(\frac34\Bigg)^{ - 4} = \frac43^4 = \frac {256}{81}</math>, <math>y = \frac {64}{27}</math>, <math>x + y = \frac {256 + 192}{81} = \frac {448}{81}</math>, yielding an answer of <math>448 + 81 = \boxed{529}</math>. | ||
== See also == | == See also == | ||
Revision as of 13:19, 17 March 2010
Problem
Suppose that 
and 
. The quantity 
can be expressed as a rational number 
, where 
and 
are relatively prime positive integers. Find 
.
Solution
We solve in general using 
instead of 
. Substituting 
, we have:
![\[x^{cx} = (cx)^x \Longrightarrow (x^x)^c = c^x\cdot x^x\]](http://latex.artofproblemsolving.com/6/d/8/6d806bc47a9f272be605a24af932316a92ed65fd.png)
Dividing by 
, we get 
.
Taking the 
th root, 
, or 
.
In the case 
, 
, 
, 
, yielding an answer of 
.
See also
| 2010 AIME I (Problems • Answer Key • Resources) | ||
| Preceded by Problem 2 |
Followed by Problem 4 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
| All AIME Problems and Solutions | ||
