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Expert: - 8/7/2009


Question

Scott,

This is a question from last year's math competition. I've looked over a few times, but I'm not sure how to go about solving it. Please help!

The arithmetic mean of a set of 78 numbers is 32. The arithmetic mean of a set of x numbers is 29. The arithmetic mean of a set of y numbers is 33. If the arithmetic mean of the (78+x+y) numbers is 31, find the ordered pair of positive integers (x,y) such that (x+y) is a minimum. Be sure to express your answer as an ordered pair in the form (x,y).

Thanks in advance!
~Ankai~

Answer
On messages that are to me, under private needs to be a, "Yes".

As it is right now, Paul Klarreich, Alon Mandes, Ahmed Salami, Abe Mantell, Socrates, Chanda Walker, eigensteve, and Vineeth Venugopal all know that I am getting some of these questions to me only.  Of course, I have no idea who these people are either.  I don't even know which country they are in, but I'm not sure they really care about questions to me.

Now lets get on with solving the problem.

Since the number of x values comes out so even and the number of y values increases the average, you want no y values.

We are given the following
Overal currnent average is 32.
Average x value is 29.
Desired average is 31

This shows that the desired average is 1 below the current average and for the values added, it is 2 above.

Since the affect of every added piece as in the same as that of having two piece there,
this means we want to add half as many pieces as we have values there.

In other words, take half of 78 and you get 39.  You want to add that mand x values and then the average will drop from 32 to 31.

Check the Work Done
----------------------------------------------------------------
Adding 39 x parts increases the number to 78 + 39 = 117.

If this needs to be checked, the current total is 78*32 = 2,496.

The total with 39 parts of type x added is 2496, for a total of 29*29 = 1131.
Now when we add 2496 and 1131, the result is 3627.

Note that the average is no 31, for if we note that currently we have 117 parts,
then (30 + 1)117 = 3,510 + 117 = 3,627.

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