2010 AMC 12A Problems/Problem 22
Contents
Problem
What is the minimum value of ?
Solution
Solution 1
If we graph each term separately, we will notice that all of the zeros occur at , where
is any integer from
to
, inclusive.
The minimum value occurs where the sum of the slopes is at a minimum, since it is easy to see that the value will be increasing on either side. That means the minimum must happen at some .
The sum of the slope at is
Now we want to minimize . The zeros occur at
and
, which means the slope is
where
.
We can now verify that both and
yield
.
Solution 2
Rewrite the given expression as follows:
Imagine the real line. For each
imagine that there are
boys standing at the coordinate
. We now need to place a girl on the real line in such a way that the sum of her distances from all the boys is minimal, and we need to compute this sum.
Note that there are boys in total. Let's label them from 1 (the only boy placed at
) to
(the last boy placed at
.
Clearly, the minimum sum is achieved if the girl's coordinate is the median of the boys' coordinates. To prove this, place the girl at the median coordinate. If you now move her in any direction by any amount , there will be
boys such that she moves
away from this boy. For each of the remaining boys, she moves at most
closer, hence the total sum of distances does not decrease.
Hence the optimal solution is to place the girl at the median coordinate. Or, more precisely, as is even, we can place her anywhere on the segment formed by boy
and boy
: by extending the previous argument, anywhere on this segment the sum of distances is the same.
By trial and error, or by solving the quadratic equation we get that boy number
is the last boy placed at
and the next boy is the one placed at
. Hence the given expression is minimized for any
.
Common part of both solutions
To find the minimum, we want to balance the expression so that it is neither top nor bottom heavy. .
Now that we know that the sum of the first 84 's is equivalent to the sum of
's 85 to 119, we can plug either
or
to find the minimum.
Note that the terms to
are negative, and the terms
to
are positive. Hence we get:
and
Hence the total sum of distances is
.
Solution 3
Since the minimum exists, we want all the s to cancel out. Thus, we want to find some
such that
Then, . The answer(expression's value) is then
, which becomes
.
See also
2010 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 21 |
Followed by Problem 23 |
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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