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.
Let the numbers be a, b, and c. We have a+b+c=t², a+b=c+z², a+c=b+y²,
and b+c=a+x². Summing the last three, we find x²+y²+z²=t², and then
a=½(y²+z²,), b=½(x²+z²), and c=½(x²+y²).
A way to find appropriate x, y, z, and t, is picking x and y randomly, and then factoring so x²+y²= t²-z²= (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.
Full solution
