2022 AMC 10B Problem 24

The rule $|f(x) - f(y)| \le \frac12 |x - y|$ says: two inputs that are $D$ apart give outputs at most $\frac{D}{2}$ apart. In graph terms, the graph is never steeper than $\frac12$ up or down.

This is a "greatest possible" question, so show that $50$ cannot be beaten, then show $50$ can happen.

Part 1: at most 50.

1. Work from the outside in. $f(f(800))$ and $f(f(400))$ are outputs for the inputs $f(800)$ and $f(400)$. So they differ by at most half of the gap between $f(800)$ and $f(400)$.
2. The gap between $f(800)$ and $f(400)$. Going straight from $400$ to $800$ only gives a bound of $\frac12 \times 400 = 200$. Use $f(300) = f(900)$ as a shortcut instead:
- $800$ and $900$ are $100$ apart, so $f(800)$ is within $50$ of $f(900)$;
- $f(900)$ equals $f(300)$;
- $300$ and $400$ are $100$ apart, so $f(300)$ is within $50$ of $f(400)$.

So $f(800)$ and $f(400)$ are at most $50 + 0 + 50 = 100$ apart.
3. Finish the bound. $f(f(800)) - f(f(400))$ is at most $\frac12 \times 100 = 50$.

Part 2: 50 is possible. Every step above must be as large as it can be: $f$ should drop by $50$ from $300$ to $400$, drop by $50$ from $800$ to $900$, and climb at slope $\frac12$ between the two inner values. Take the graph that joins these points with straight segments, and is flat to the left of $-50$ and to the right of $900$:

4. Check the steepness. The slopes of the pieces are $\frac{50}{100} = \frac12$, $\frac{-25}{250} = -\frac{1}{10}$, $\frac{-50}{100} = -\frac12$, $\frac{100}{400} = \frac14$ and $\frac{-50}{100} = -\frac12$. None is steeper than $\frac12$, so the whole graph follows the rule.
5. Check the conditions. $f(300) = f(900) = 0$. Then $f(800) = 50$ and $f(400) = -50$, so
$$f(f(800)) - f(f(400)) = f(50) - f(-50) = 25 - (-25) = 50.$$

The answer is $\boxed{\textbf{(B)}\ 50}$.

The rule means $f$ can at most halve distances, and applying $f$ twice halves again. The hidden key is the equal pair $f(300) = f(900)$: it gives a shortcut from $400$ to $800$ of total length $100$ instead of $400$. For a "greatest possible" answer you then need an example, and a connect-the-dots graph with slopes no steeper than $\frac12$ is the easiest way to make every step as large as allowed.