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.
Solution 3 (Essentially the same thing as Solution 1)
So we know that are integers so we can use this to our advantage
Using , we get the equation
and
where
is a decimal digit placeholder. (Ex.
provides the value
)
Attempting to solve for using a system of equations, we get
Since we know that and
are both integers, we know that
and by extension
Attempting to solve for again using the system
where
is another decimal digit placeholder,
gives us
This leads to
Plugging in the values of and
into
, we get
Substituting the values of , we get
and
: We can say that
and
because we are given that
and
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 |