The Cyclic Sum

The problem

Let $a, b, c, d$ be positive real numbers. Consider the sum

$S = \frac{a}{b} + \frac{b}{c} + \frac{c}{d} + \frac{d}{a}.$

What is the smallest possible value of $S$, and why can it not be smaller?

Before you begin

First guess the answer by trying values — that part is quick. Then notice something honest: guessing the minimum and proving the minimum are two very different acts. The proof is where the machinery walks in.

1. What happens when all four numbers are equal?
2. Multiply the four terms together. What is their product?
3. Which named inequality connects the average of positive numbers to their product?

Dissect the task

1. Here is the distinguishing mark of the Olympiad tier: this problem cannot be reached by structure-spotting alone. What piece of technical equipment does a child need before it is even fair to pose?
2. The first two worksheets were elementary but hard. This one is technical and hard. In your own school, when would you introduce the machinery, and to whom?