⎩ ⎪ ⎪ ⎨ ⎪ ⎪ ⎧ x 2 + x y + y 2 y 2 + y z + z 2 z 2 + z x + x 2 = = = 1 9 6 7 7 9
Considering the system of Diophantine equations above, what is the value of x y + y z + z x ?
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Note that ( 2 1 5 , 2 1 1 7 , − 2 1 4 3 ) and ( − 2 1 5 , − 2 1 1 7 , 2 1 4 3 ) are also solutions, which lead to x y + x z + y z = − 4 1 . The problem should say that only integer solutions are desired.
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Sorry, I wrote Diophantine equations before. It was probably erased after edition.
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From the first equation, x 2 + x y + y 2 = 1 9 , when we solve for x in terms of y, it can be calculated with a quadratic formula:
x = 2 − y ± y 2 − 4 ( y 2 − 1 9 ) = 2 − y ± 7 6 − 3 y 2
Considering the root of the quadratic discriminant, 7 6 − 3 y 2 > 0. Hence, y ∈ [-5 , 5].
Since this system is of Diophantine equations, the solutions of all variables are integers. Therefore, when testing for y-integers, we will get:
y = -5 → x = 2 5 ± 7 6 − 3 ( 5 2 ) = 2 5 ± 1 = 2 or 3.
y = -4 → x = 2 4 ± 7 6 − 3 ( 4 2 ) = 2 4 ± 2 8 (not integers)
y = -3 → x = 2 3 ± 7 6 − 3 ( 3 2 ) = 2 3 ± 7 = -2 or 5.
y = -2 → x = 2 2 ± 7 6 − 3 ( 2 2 ) = 2 2 ± 8 = -3 or 5.
y = -1 → x = 2 1 ± 7 6 − 3 ( 1 2 ) = 2 1 ± 7 3 (not integers)
y = 0 → x = 2 0 ± 7 6 − 3 ( 0 2 ) = 2 7 6 (not integers)
y = 1 → x = 2 − 1 ± 7 6 − 3 ( 1 2 ) = 2 − 1 ± 7 3 (not integers)
y = 2 → x = 2 − 2 ± 7 6 − 3 ( 2 2 ) = 2 − 2 ± 8 = 3 or -5.
y = 3 → x = 2 − 3 ± 7 6 − 3 ( 3 2 ) = 2 − 3 ± 7 = 2 or -5.
y = 4 → x = 2 − 4 ± 7 6 − 3 ( 4 2 ) = 2 − 4 ± 2 8 (not integers)
y = 5 → x = 2 − 5 ± 7 6 − 3 ( 5 2 ) = 2 − 5 ± 1 = -2 or -3.
As a result, the possible pairs of (x , y) include:
{(2 , -5), (2 , 3), (3 , -5), (3 , 2), (5 , -2), (5 , -3), (-2 , -3), (-2 , 5), (-3 , -2), (-3 , 5), (-5 , 2), (-5 , 3)}
Similarly, when we evaluate the second equation, the possible pairs of (y , z) include:
{(2 , 7), (2 , -9), (7 , 2), (7 , -9), (9 , -2), (9 , -7), (-2 , 9), (-2 , -7), (-7 , 9), (-7 , -2), (-9 , 2), (-9 , 7)}
Finally, for the third equation, the possible pairs of (z , x) include:
{(3 , 7), (3 , -10), (7 , 3), (7 , -10), (10 , -3), (10 , -7), (-3 , 10), (-3 , -7), (-7 , 10), (-7 , -3), (-10 , 7), (-10 , 3)}
Now the only applicable triples (x , y , z) corresponding to those integers include: (3 , 2 , 7) or (-3 , -2 , -7).
Either way, the value of x y + y z + z x = 6 + 14 + 21 = 41.