(a) State Brouwer's fixed point theorem in the plane.

(b) Let $\mathbf{a}, \mathbf{b}, \mathbf{c}$ be unit vectors in $\mathbb{R}^{2}$ making $120^{\circ}$ angles with one another. Let $T$ be the triangle with vertices given by the points $\mathbf{a}, \mathbf{b}$ and $\mathbf{c}$ and let $I, J, K$ be the three sides of $T$. Prove that the following two statements are equivalent:

(1) There exists no continuous function $f: T \rightarrow \partial T$ with $f(I) \subseteq I, f(J) \subseteq J$ and $f(K) \subseteq K$.

(2) If $A, B, C$ are closed subsets of $\mathbb{R}^{2}$ such that $T \subseteq A \cup B \cup C, I \subseteq A, J \subseteq B$ and $K \subseteq C$, then $A \cap B \cap C \neq \emptyset$.

(c) Let $f, g: \mathbb{R}^{2} \rightarrow \mathbb{R}$ be continuous positive functions. Show that the system of equations

$$\begin{aligned} &\left(1-x^{2}\right) f^{2}(x, y)-x^{2} g^{2}(x, y)=0 \\ &\left(1-y^{2}\right) g^{2}(x, y)-y^{2} f^{2}(x, y)=0 \end{aligned}$$

has four distinct solutions on the unit circle $\mathbb{S}^{1}=\left\{(x, y) \in \mathbb{R}^{2}: x^{2}+y^{2}=1\right\}$.