The following simple-looking inequality for complex numbers in the unit disk generalizes Problem B5 on the Putnam contest 2020:

Theorem 1.Let $z_1, z_2, \ldots, z_n$ be $n$ complex numbers such that $\left|z_i\right| \leq 1$ for each $i \in \left\{1,2,\ldots,n\right\}$. Prove that \begin{align} \left| z_{1} + z_{2} + \cdots + z_n - n \right| \geq \left| z_{1} z_{2} \cdots z_n - 1 \right| , \end{align} and equality holds only if at least $n-1$ of the $n$ numbers $z_1, z_2, \ldots, z_n$ equal $1$.

In the particular case when $n = 4$, the theorem can be proved using stereographic projection onto a line, followed by a longish computation. This is how both proposed solutions go.

On the other hand, in the general case, the only elementary solution I know was given by @mela_20-15 on AoPS (spread over several posts). It has some beautiful parts (Cauchy induction), but also some messy ones (tweaking the points to lie on the unit circle in the induction step). There might also be a heavily analysis-based proof in Kiran Kedlaya's solutions (not sure if Theorem 1 is proved in full there).

Question.What is the "proof from the book" for Theorem 1?

Someone suggested to me to try to interpolate expressions of the form $\dbinom{n-1}{k-1}^{-1} \left|\sum\limits_{i_1 < i_2 < \cdots < i_k} z_{i_1} z_{i_2} \cdots z_{i_k} - \dbinom{n}{k}\right|$ between the left and the right hand sides in Theorem 1; but this does not work. For example, the inequality $\dfrac{1}{2} \left| z_1 z_2 + z_2 z_3 + z_1 z_3 - 3\right| \geq \left|z_1 z_2 z_3 - 1\right|$ fails quite often even on the unit circle.

A **warning**: Inequalities like Theorem 1 are rather hard to check numerically. Choosing the $z_i$ uniformly will rarely hit close to the equality case; usually the left hand side will be much larger than the right. Near the equality case, on the other hand, it is hard to tell whether the answer comes out right legitimately or whether accumulated errors have flipped the sign.