2006 AIME I Problems/Problem 5
Contents
[hide]Problem
The number can be written as where and are positive integers. Find .
Solution 1
We begin by equating the two expressions:
Squaring both sides yields:
Since , , and are integers, we can match coefficients:
Solving the first three equations gives:
Multiplying these equations gives .
Solution 2
We realize that the quantity under the largest radical is a perfect square and attempt to rewrite the radicand as a square. Start by setting , , and . Since
we attempt to rewrite the radicand in this form:
Factoring, we see that , , and . Setting , , and , we see that
so our numbers check. Thus . Square rooting gives us and our answer is
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 | ||
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