2011 AMC 12A Problems/Problem 20
Contents
Problem
Let , where , , and are integers. Suppose that , , , for some integer . What is ?
Solution 1
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 .
Solution 2
is some non-monic quadratic with a root at . Knowing this, we'll forget their silly , , and and instead write it as .
, so is a multiple of 6. They say is between 50 and 60, exclusive. Notice that the only multiple of 6 in that range is 54. Thus, .
, so is a multiple of 7. They say is between 70 and 80, exclusive. Notice that the only multiple of 7 in that range is 77. Thus, .
Now, we solve a system of equations in two variables.
See also
2011 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 19 |
Followed by Problem 21 |
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All AMC 12 Problems and Solutions |
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