Difference between revisions of "2006 AIME I Problems/Problem 5"
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<math> abc = \sqrt{52 \cdot 234 \cdot 72} = 936</math> | <math> abc = \sqrt{52 \cdot 234 \cdot 72} = 936</math> | ||
+ | |||
+ | ALTERNATIVELY: | ||
+ | |||
+ | Since this is the AIME and you do not have a calculator solving <math> abc = \sqrt{52 \cdot 234 \cdot 72} = 936</math> might prove difficult. | ||
+ | So instead use the three equations given above. | ||
+ | |||
+ | <math> ab = 52 </math> | ||
+ | |||
+ | <math> ac = 234 </math> | ||
+ | |||
+ | <math> bc = 72 </math> | ||
+ | |||
+ | Thus <math> a = 52/b = 234/c </math> | ||
+ | |||
+ | <math> 52c = 234b </math> | ||
+ | |||
+ | <math> c = 234/52b </math> | ||
+ | |||
+ | <math> c = 9/2b </math> | ||
+ | |||
+ | Plugging into last equation leads to: | ||
+ | |||
+ | <math> 9/2b^2 = 72 </math> | ||
+ | |||
+ | <math> b = 4 </math> | ||
+ | |||
+ | Plugging into others you get | ||
+ | |||
+ | <math>a=13</math> | ||
+ | |||
+ | <math>b=4</math> | ||
+ | |||
+ | <math>c=18</math> | ||
+ | |||
+ | Much easier than taking crazy square roots without a calculator! | ||
If it was required to solve for each variable, dividing the product of the three variables by the product of any two variables would yield the third variable. Doing so yields: | If it was required to solve for each variable, dividing the product of the three variables by the product of any two variables would yield the third variable. Doing so yields: |
Revision as of 00:43, 23 November 2007
Problem
The number can be written as
where
and
are positive integers. Find
.
Solution
We begin by equating the two expressions:
Squaring both sides yeilds:
Since ,
, and
are integers:
1:
2:
3:
4:
Solving the first three equations gives:
Multiplying these equations gives:
ALTERNATIVELY:
Since this is the AIME and you do not have a calculator solving might prove difficult.
So instead use the three equations given above.
Thus
Plugging into last equation leads to:
Plugging into others you get
Much easier than taking crazy square roots without a calculator!
If it was required to solve for each variable, dividing the product of the three variables by the product of any two variables would yield the third variable. Doing so yields:
Which clearly fits the fourth equation:
See also
2006 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 4 |
Followed by Problem 6 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |