Problem. (AoPS) For any positive integers $k,m$, let $S_k(m)$ be the number $n\in {1,…,2^k}$ which can be expressed as $$n = \sum_{i=0}^k \varepsilon_i\cdot 2^i\qquad \varepsilon_i\in {-1,0,1}$$ in exactly $m$ different ways. Show that $S_k(m) = \varphi(m)$ for all sufficiently large $k$.

Remarks. Straight up, I basically had no idea how something like combinations of a sum can be related to $\varphi(m)$ at all (that’s part of the appeal I guess).

So what I should have started off with is to list all the small distributions (i.e. find $S_k$ for small $k$), but alas, I’m lazy and have all the time in the world.

Problem. (239MO 2019 11.7) Given positive $a_1,\ldots,a_n,b_1,\ldots,b_n,c_1,\ldots,c_n$. Let $m_k$ be the maximum product $a_ib_jc_l$ for triples $(i,j,l)$ satisfying $\max(i,j,l)=k$. Prove that $$ (a_1+\dots+a_n)(b_1+\dots+b_n)(c_1+\dots+c_n)\leqslant n^2(m_1+\dots+m_n).$$

Problems from the 239 Olympiad, held by Lyceum 239 in St Petersburg Russia for the year 2019.

Note: blue text is broken atm after porting over to Hugo site.

So a while ago, Eugene had the idea of having a “bluetext” walkthrough. The idea was that you have the walkthrough alongside a complete solution so you have both the thought process and how it would translate on paper.

I tried to compile a collection of bluetexted problems but never really got around to it. Anyway, here’s the very first one I tried.