Each of a, b, c, d, e and f is a positive integer satisfying this system of equations:

a^-2 + b^-2 + c^-2 + d^-2= e^-2, and:

1/a+ 1/b+ 1/c+ 1/d= f^-2

Determine the smallest value of f. What are the next three smallest values of f?

let a=b=c=d
then we have
1/a+1/b+1/c+1/d=4/a
thus we want
4/a=1/f^2
a=4f^2
and we also need
1/a^2+1/b^2+1/c^2+1/d^2=4/a^2=1/e^2
a^2=4e^2  since a,e>0
a=2e
e=a/2=(4f^2)/2=2f^2
thus for any postive f we can find a,b,c,d,e satifying the equations.
thus the smallest value of f is 1, and the next 3 are 2,3,4.

Posted by Daniel on 2011-09-09 10:54:06