2013 AMC 12A Problems/Problem 16
, , are three piles of rocks. The mean weight of the rocks in is pounds, the mean weight of the rocks in is pounds, the mean weight of the rocks in the combined piles and is pounds, and the mean weight of the rocks in the combined piles and is pounds. What is the greatest possible integer value for the mean in pounds of the rocks in the combined piles and ?
Let pile have rocks, and so on.
The mean weight of and together is , so the total weight of and is
To get the total weight of and , we need to add the total weight of and subtract the total weight of
And then dividing by the number of rocks and together, to get the mean of and ,
Now, to get rid of the in the numerator, we use two definitions of the total weight of and
Substituting back in,
Note that , and the maximal value of this factor occurs when
Also note that must cancel to give an integer value, and the only fraction that satisfies both these conditions is
Plugging in, we get
Suppose there are rocks in the three piles, and that the mean of pile C is , and that the mean of the combination of and is . We are going to maximize , subject to the following conditions:
which can be rearranged as:
Let us test is possible. If so, it is already the answer. If not, there will be some contradiction. So the third equation becomes
So , , , therefore,
, which gives us a consistent solution. Therefore is the answer.
(Note: To further illustrate the idea, let us look at and see what happens. We then get , which is a contradiction!)
Obtain the 3 equations as in solution 2.
Combining the 1st and 2nd equations, we see that
Subtracting equation 3 from equation 2, we have
In order for the coefficients to be positive,
Thus, the greatest integer value is , choice .
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