Difference between revisions of "1955 AHSME Problems/Problem 38"
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Latest revision as of 23:35, 28 August 2020
Problem 38
Four positive integers are given. Select any three of these integers, find their arithmetic average,
and add this result to the fourth integer. Thus the numbers 
, and 
are obtained. One of the original integers is:
Solution
Define numbers 
and 
to be the four numbers. In order to satisfy the following conditions, the system of equation should be constructed. (It doesn't matter which variable is which.)
![\[\frac{a+b+c}{3}+d=29\]](http://latex.artofproblemsolving.com/d/8/c/d8c28a3df93fae35eb1494d34fa4434f2732c31a.png)
![\[\frac{a+b+d}{3}+c=23\]](http://latex.artofproblemsolving.com/4/3/8/438bca8582b1a3151fb9556cfe85b1d7c9d242b8.png)
![\[\frac{a+c+d}{3}+b=21\]](http://latex.artofproblemsolving.com/c/3/c/c3c1fdba5267c9a0c2f665276ff0c9058d32707e.png)
![\[\frac{b+c+d}{3}+a=17\]](http://latex.artofproblemsolving.com/3/6/f/36f018fb5eb5323a46a473e72de93d1a135a8ce3.png)
Adding all of the equations together, we get: 
. This means that 
.
We can determine that 
. This, with some algebra, means that 
. must be 
.
See Also
In order to see the rest of the problem set, click here
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
