SIMO Retiree Blog

(USATST '25 P2) Let $a_1,a_2,\dots,$ and $b_1,b_2,\dots,$ be sequences of real numbers for which $a_1 > b_1$ and $a_{n+1} = a_n^2-2b_n, b_{n+1}=b_n^2-2a_n$ for all positive integers $n$. Prove that $a_1,a_2,\dots,$ is eventually increasing (i.e. there exists a positive integer $N$ for which $a_k < a_{k+1}$ for all $k>N$). (Dylan here.) I will be writing this post as I am trying to solve the above problem.

(Drew here.) I've spent a lot of time browsing random math pages on Wikipedia  (blue link). Sometimes I find things which are actually useful in olympiads, but a lot of the time there's this void of open problems. Of these open problems, some were only solved recently, and they can be used to blast olympiad problems.

(Andrew here.)  In research, it often happens that people rediscover results lost to the sands of time, which always makes me wonder what gemstones have been lost and are waiting to be rediscovered. It also means that a lot of surprising things can be learned from reading old papers.

(This is Glen.) I was sorting through my stuff and found a piece of paper from when I tried last year's NTST problems, which I presumably saved because I thought I might eventually want to write a post about it. So here I am, writing a post so I can safely dump this piece of paper into my recycling bin.