Product of distances from a point in $\mathbb{C}$, Stein-Shakarchi I 'm trying to do this exercise from Stein - Shakarchi. 
Let $w_{1} \ldots w_{n}$ be points on the unit circle in $\mathbb{C}$. Prove that there exist a point $z$ on the unit circle such that the product of the distances from $z$ to the points $w_{j}$ , $1 \leq j \leq n$ is at least $1$. 
Conclude that there exist a point $w$ on the unit circle such that the product of the distances from $w$ to the points $w_{j}$ , $1 \leq j \leq n$ is exactly equal to $1$.
Any hint ?
 A: Note that the second statement follows from the first by the intermediate value theorem. Let $f(p)$ be the product of the distances from the given points to the point $p$. Since $f(p)$ can be zero (when $p=w_1$) and at least $1$, there must be some point as $p$ moves around the circle where $f(p)=1$. 
Now consider the holomorphic function
$$g(z)=\prod_i (z-w_i).$$
By the maximum principle, the maximum modulus of of this function on the unit disk is attained on the boundary (the unit circle). But setting $z=0$, we see that the maximum modulus is at least $1$.
Alternatively, you can show that the integral of $g$ around the unit circle has modulus $1$, so the integral of $|g(z)|$ must be at least $1$ by the triangle inequality. Then $g|(z)|$ must be at least $1$ at some point because it is continuous. 
A: Potato has already given a correct answer. I just want to add something.
Setting 
$$
g(z)=\prod_{j=1}^n (z-w_j),
$$
we obtain a non-constant entire analytic function with the property that 
$$
|g(0)|=\prod_{j=1}^n |w_j|=1,
$$
and using the Maximum Modulus Principle not only we obtain that that
$$
|g(0)|\le \max_{|z|=1}|g(z)|,
$$
but we obtain A LITTLE MORE; namely
$$
|g(0)|< \max_{|z|=1}|g(z)|,
$$
as $g$ is non-constant! Hence there is a $w$, $|w^*|=1$, such that $|g(w^*)|>1$.
This second question is a consequence of the continuity of $G(z)=|g(z)|$ is the unit circle $C$, since $G(w_j)=0$ and $G(w^*)>1$ - Note that continuity of $G$ on $C$ implies that $G[C]$ is a closed interval.
A: Yes, the hint is: maximum principle.
