Question

If the sum of 3 non-zero distinct real numbers a, b, and c is 2, and the two sets {a,b,c} and {1/a,1/b,1/c} are the same, what is the value of a^2+b^2+c^2?

Note: Two sets are the same if there is a one-to-one correspondence between their elements. For example, the sets {1,2,3} and {3,2,1} are the same. Neither of them are the same as {1,2,1}.

Answer 1

assuming a<=b<=c
a = 1/c
b = 1/b
c = 1/a
a+b+c=2
We see that b=±1, so a+c=1 or 3

If b=1,
a+ 1/a = 1
a^2 - a + 1 = 0
2a = 1±√3 i
Nope

If b = -1,
a + 1/a = 3
a^2 - 3a + 1 = 0
a = (3±√5)/2
c = 2/(3±√5) = (3∓√5)/2

a^2+b^2+c^2 = (14+6√5)/4 + 1 + (14-6√5)/4 = 8