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Nonnegative Integer Decision Poser (Posted on 2016-03-22) Difficulty: 3 of 5
Consider this system of equations:
A+B*C = (A+B)2, and:
B+C*A = (B+C)2, and:
C+A*B = (C+A)2, and:

It is observed that A=B=C=0 trivially satisfies the above system of equations.

Does there exist any further nonnegative integer solutions to the above system of equations?

Provide adequate reasoning for your answer.

No Solution Yet Submitted by K Sengupta    
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Solution Solution Comment 1 of 1
First, look for any other solutions with 0.

If A=0 then the system becomes:
B*C = B^2
B = (B+C)^2
C = C^2

C is either 0 or 1.  If C=0 then B=0, yielding the trivial solution (0,0,0).  If C=1 then B = B^2 = (B+1)^2, which has no solutions.

Then the only solution (A,B,C) with at least one zero is (0,0,0).

If A=B then the original system reduces to: 
A*(1+C) = (2A)^2
A*(1+C) = (A+C)^2
C+A^2 = (A+C)^2

The first two imply 2A = A+C, or A=C.  Then A=B=C, which implies they all equal 1/3 or 0, but 1/3 is not integral.

To try to find remaining nonnegative integer solutions assume without loss of generality 1<=A<B<C.

The second equation states B + C*A = (B+C)^2 = B^2 + 2BC + C^2.  But from the inequality B<B^2 and A*C<C^2.  Therefore B + C*A < B^2 + 2BC + C^2 and the equality has no solutions.

The only nonnegative integer solution is (0,0,0).

  Posted by Brian Smith on 2016-03-22 13:54:30
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