Difference between revisions of "2006 Cyprus MO/Lyceum/Problem 5"
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<math>a/b=1</math> doesn't work, since <math>a</math> and <math>b</math> are defined to be greater than <math>1</math>. The next smallest thing we try is <math>a/b=2</math>. | <math>a/b=1</math> doesn't work, since <math>a</math> and <math>b</math> are defined to be greater than <math>1</math>. The next smallest thing we try is <math>a/b=2</math>. | ||
| − | This gives <math>b=(a/b)^7=2^7=128</math>, so <math>a=2b=2(128)=256</math>. Thus, our sum is <math>128+256=\ | + | This gives <math>b=(a/b)^7=2^7=128</math>, so <math>a=2b=2(128)=256</math>. Thus, our sum is <math>128+256=384</math>, and the answer is <math>\mathrm{(A)}</math>. |
==See also== | ==See also== | ||
Latest revision as of 10:03, 27 April 2008
Problem
If both integers 
are bigger than 1 and satisfy 
, then the minimum value of 
is
Solution
Since 
is greater than 
and therefore not equal to zero, we can divide both sides of the equation by 
to obtain 
, or
![\[\left(\frac{a}{b}\right)^7=b\]](http://latex.artofproblemsolving.com/8/c/3/8c37d410a9ea0f6c3660316e40b39b28fc1d5a88.png)
Since is an integer, we must have 
is an integer. So, we can start testing out seventh powers of integers.
doesn't work, since 
and are defined to be greater than . The next smallest thing we try is 
.
This gives 
, so 
. Thus, our sum is 
, and the answer is 
.
See also
| 2006 Cyprus MO, Lyceum (Problems) | ||
| Preceded by Problem 4 |
Followed by Problem 6 | |
| 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 | ||
