Difference between revisions of "2011 AMC 12A Problems/Problem 20"
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Revision as of 11:14, 19 February 2011
Problem
Let , where
,
, and
are integers. Suppose that
,
,
,
for some integer
. What is
?
Solution
From , we know that
.
From the first inequality, we get . Subtracting
from this gives us
, and thus
. Since
must be an integer, it follows that
.
Similarly, from the second inequality, we get . Again subtracting
from this gives us
, or
. It follows from this that
.
We now have a system of three equations: ,
, and
. Solving gives us
and from this we find that
Since , we find that
.
See also
Pretty esay, right? =)
2011 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 19 |
Followed by Problem 21 |
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 |