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Classical Rules 1 (Posted on 2006-04-17) Difficulty: 3 of 5
Find three positive rational numbers such that their sum is a square, and the sum of any pair exceeds the third by a square.

Classical Rules: Let a "square" be any number that is the square of a rational number.

No Solution Yet Submitted by goFish    
Rating: 3.5000 (4 votes)

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Solution Full solution | Comment 4 of 10 |
Let the numbers be a, b, and c. We have a+b+c=t, a+b=c+z, a+c=b+y, and b+c=a+x. Summing the last three, we find x+y+z=t, and then a=(y+z,), b=(x+z), and c=(x+y).

A way to find appropriate x, y, z, and t, is picking x and y randomly, and then factoring so x+y= t-z= (t+z)(t-z).

An example: I picked x=4 and y=7; then, (t+z)(t-z)=63, so I can choose t=8 and z=1, finally leading to a=25, b=17/2, and c=65/2.


  Posted by Federico Kereki on 2006-04-17 14:59:53
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