Difference between revisions of "2011 AMC 12A Problems/Problem 21"
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Revision as of 21:52, 3 July 2013
Problem
Let , and for integers
, let
. If
is the largest value of
for which the domain of
is nonempty, the domain of
is
. What is
?
Solution
The domain of is defined when
.
. Applying the domain of
and the fact that square roots must be positive, we get
. Simplify this to arrive at the domain for
, which is defined when
. Repeat this process for
to get a domain of
. For
, since square roots are positive, we can exclude the negative values of the previous domain to arrive at
as the domain of
. We now arrive at a domain with a single number that defines
, however, since we are looking for the largest value for
for which the domain of
is nonempty, we must continue until we arrive at a domain that is empty. We continue with
to get a domain of
. Solve for
to get
. Since square roots cannot be negative, this is the last nonempty domain. We add to get
.
See also
2011 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 20 |
Followed by Problem 22 |
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 | |
All AMC 12 Problems and Solutions |
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