This problem is very strange. I am embarrassed that not only can I not do this problem but I do not see any errors in my work.
Here goes:
a1+b1+c1=abc4+abc4+abc4=abc12=21soabc=24
Since
ab=4, we get
c=6 But
ac=4 so
b=6 (similarly we get
a=6). But this contradicts
ab=ac=bc=4
2nd method.
abc=24 Now
bcab=ca=1 So
a=c Same argument that
b=c so
a=b=c. This yields that
a=b=c=2 (assuming positive numbers). Then we get
21+21+21=21 which is also not true.
Now if a, b and c were all -2, then the sum would not be 1/2. If only one was -2 and the others both 2, than that would contradict that the product of any 2 would be 4.
What is going on here?