(Alibaba Contest Finals 2020/18) Let $a_1,a_2,…,a_n$ be positive reals whose squares sum to 1. Show that there exists a choice of $\varepsilon_i\in{-1,1}$ such that $$\left|\sum_{i=1}^n \varepsilon_i a_i\right| \le \frac{1}{a_1+a_2+…+a_n}$$

While it’s still fresh in my mind I guess.

(Putnam 1999 B6) Let $S$ be a finite site of positive integers. Suppose for any positive integer $n$, $(n,s)$ is either 1 or $s$. Show that there exists $s,t\in S$ such that $(s,t)$ is prime.

Remarks. I never understood how to figure out the extremal solution for this. Like, what are you supposed to do, just guess?

Walkthrough. Of course, the first thing to try is $|S| =1$. It is obvious that if $(n,s) = 1$ or $s$ always, then $s$ itself has to be prime! But which $n$ tells us that? We can think of it by “varying $n$” - $1=n_0 \mid n_1 \mid … \mid n_k$, and of course at each step we add in a prime factor so that $(n,s)$ varies by a prime factor, but somehow the conditions dictate that it must jump from 1 to $s$. So on the very first prime factor it must jump to $s$ which means that $s$ itself is prime.

Because I’m silly and didn’t set up Twitch to record our livestream, here’s roughly our thought processes for the three problems. (Solved with Eugene Lee on stream.)

Some 3am ramblings for my IMO guest-session students (back in 2019).