VWO 1991

(a) Show that for every $n\in\mathbb{N}$ there is exactly one $x\in\mathbb{R}^+$ so that $x^n+x^{n+1}=1$ . Call this $x_n$ . (b) Find $\lim\limits_{n\rightarrow+\infty}x_n$ .

Given $\Delta ABC$ equilateral, with $X\in[A,B]$ . Then we define unique points Y,Z so that $Y\in[B,C]$ , $Z\in[A,C]$ , $\Delta XYZ$ equilateral.

If $Area\left(\Delta ABC\right) = 2 \cdot Area\left(\Delta XYZ\right)$ , find the ratio of $\frac{AX}{XB},\frac{BY}{YC},\frac{CZ}{ZA}$ .