Proposed solution of problem J427

Dear Editor, I would like to submit the following solution of problem J427

Find all complex numbers which satisfy simultaneously the equations:

x + y + z = 1, x^{3} + y^{3} + z^{3} = 1, x^{2} + 2yz = 4

Proof

x^{3} + y^{3} + (1 − x − y)^{3} = 1 ⟺ (x + y)(x + y − 1 − xy) = 0 (1)

If y = −x we have z = 1 and from the third equation we get

x^{2} − 2x − 4 = 0 ⟺ x = 1 ± i\sqrt{5}

so here the solutions are

(x, y, z) = (1 − i\sqrt{5}, −1 + i\sqrt{5}, 1), (1 + i\sqrt{5}, −1 − i\sqrt{5}, 1)

From (1) we have also

x + y + 1 − xy = 0 ⟺ x(1 − y) = 1 − y  (2)

If y = 1 we get x = −z and from the third equation we have

x^{2} − 2x = 4 ⟺ x = 1 ± i\sqrt{5}

so the solutions we get are

(x, y, z) = (1 + i\sqrt{5}, 1, −1 − i\sqrt{5}), (1 − i\sqrt{5}, 1, −1 + i\sqrt{5})

If in (2) we take x = 1 we get y = −z and from the third equation

1 − 2y^{2} = 4 ⟺ y = ±i\sqrt{3/2}

so the solutions are

(x, y, z) = (1, i\sqrt{\frac{3}{2}}, −i\sqrt{\frac{3}{2}}) , ((1, −i\sqrt{\frac{3}{2}}, i\sqrt{\frac{3}{2}})

Roma 10/14/2017 Best regards Paolo Perfetti

Perfetti Paolo, dipartimento di matematica, Università degli studi di Tor Vergata Roma, via della ricerca scientifica, 00133 Roma, Italy – email: perfetti@mat.uniroma2.it