Moving Points

Philosophy

The idea of the unit (very generally) is the following: We first rewrite the problem in terms of a few fixed points and a point $Q$ moving over a path $\mathcal{P}$. We show that if we move $Q$ over this path $\mathcal{P}$ and the problem is true for $k$ cases of $Q$, then it is true for all $Q$ on $\mathcal{P}$. We then find $k$ cases of $Q$ for which we can solve the problem and finish.

For example take 2010 IMO 2, one of the first problems in the unit. We fix $A,B,C,I,D$ and move $E$ over the path $(ABC)$, and define $F,G$ based on $E$. Moving Points allows us to show that we only need to solve 2010 IMO 2 for $3$ cases of $E$, which is easy to do ($E\equiv B,D,C$ for example).

An extended warning

Doing the above requires a lot of theory: There are about 16 pages of theory in the unit and Evan says that if he is your instructor, you should understand as much as you can before the zoom call. With all this theory, it is also easy to mess up, and not easy to find any mistakes either (Evan probably won't find your mistakes in the check off process either). The fail rate for moving points on USA team selection tests are >50%, occasionally close to 90%.

"Please dont" - Evan Chen

Succinctly, in the words of another editor:

"Moving points: it moves your points from 7 to 0" -- Neil Kolekar

However if you have a strong geo background, and you're willing to sell your soul in some future competition for 7 points, then feel free to learn how to cheese on proof-based competitions and get away with it (unless you mess up).

Notable Problems

1. 2017 RMM 6, the last problem in the unit and the only 9 pointer as of the time of writing (Dec 4 2022).

2. Mini Survey, in contrast the problem with the least amount of clubs, as well as the most enjoyable problem.

Trivia

"sacrilegious" -- Eric Shen