Difference between revisions of "2006 AIME I Problems/Problem 5"
m (2006 AIME I Problem 5 moved to 2006 AIME I Problems/Problem 5) |
(adding a note as to the method used at the beginning of the problem) |
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== Solution == | == Solution == | ||
| + | We begin by [[equate | equating]] the two expressions: | ||
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<math> a\sqrt{2}+b\sqrt{3}+c\sqrt{5} = \sqrt{104\sqrt{6}+468\sqrt{10}+144\sqrt{15}+2006}</math> | <math> a\sqrt{2}+b\sqrt{3}+c\sqrt{5} = \sqrt{104\sqrt{6}+468\sqrt{10}+144\sqrt{15}+2006}</math> | ||
Revision as of 02:26, 12 July 2006
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:
If it was required to solve for each variable, dividing the product of the three variables by the product of any two variables would yeild the third variable. Doing so yeilds:
Which clearly fits the fourth equation:
