IMC 2000 · Day 1 · P2

Let $$P(x,y) = \frac{p(x) - p(y)}{x - y} = x^4 + x^3 y + x^2 y^2 + x y^3 + y^4 + 1$$ and $$Q(x,y) = \frac{q(x) - q(y)}{x - y} = x^4 + x^3 y + x^2 y^2 + x y^3 + y^4 + x + y.$$ We need those pairs $(w,z)$ which satisfy $P(w,z) = Q(w,z) = 0$.

From $P - Q = 0$ we have $w + z = 1$. Let $c = wz$. After a short calculation we obtain $c^2 - 3c + 2 = 0$, which has the solutions $c = 1$ and $c = 2$. From the system $w + z = 1$, $wz = c$ we obtain the following pairs: $$\left( \frac{1 \pm \sqrt{3} i}{2}, \frac{1 \mp \sqrt{3} i}{2} \right) \quad \text{and} \quad \left( \frac{1 \pm \sqrt{7} i}{2}, \frac{1 \mp \sqrt{7} i}{2} \right).$$