SIMO Retiree Blog

(This is Yan Sheng.) Last time we studied a family of inequalities involving absolute values on linear polynomials, and asked the question of how lazy we can be to prove inequalities of that form. This week, we will be applying the same approach to a different family: 3-variable homogeneous symmetric inequalities of low degree. A polynomial $P(x,y,z)$ in 3 variables is called homogeneous of degree $d$ if each of its terms is of degree $d$, and symmetric if $P(x,y,z)=P(x',y',z')$ for all permutations $x',y',z'$ of $x,y,z$. Inequalities involving homogeneous symmetric polynomials include the AM-GM inequality$$\left(\frac{x+y+z}3\right)^3\ge xyz,$$and the Schur inequality$$x^r(x-y)(x-z)+y^r(y-x)(y-z)+z^r(z-x)(z-y)\ge0$$for integers $r\ge0$. For the rest of this post, write $\mathcal P^+_d$ for the family of all homogeneous symmetric polynomials $P(x,y,z)$ of degree $d$ such that $P(x,y,z)\ge0$ holds for all $x,y,z\ge0$. Our Main Problem is to describe $\math...

(This is Yan Sheng.) What does it mean to "properly understand" some mathematical result? For me, I find it the most satisfying when I can answer the following two questions: What is the minimal set of special cases that I need to verify to prove it? How could I have come up with it myself? In this and the next blog post, I'll describe two different situations in olympiad inequalities that I've tried to understand better recently, by answering the two questions above. Theorem (Popoviciu 1965): Let $f:[a,b]\to\mathbb R$ be a convex function. Then for any $x,y,z\in[a,b]$, we have$$\begin{align*}&f(x)+f(y)+f(z)+3f\left(\frac{x+y+z}3\right)\\&\ge2\left(f\left(\frac{x+y}2\right)+\left(\frac{y+z}2\right)+\left(\frac{z+x}2\right)\right).\end{align*}$$ What an interesting statement! It's not immediately clear how to prove it with Jensen's inequality, and it makes me wonder what other similar inequalities hold for convex functions. Let's try provin...

This is Glen again. There's been a lack of regular olympiad content recently, and since no one posted this week I thought I'd attempt a problem and write about my thought process. There was some discussion recently on the retirees' Discord server about APMO, so I figured I'd look at the most recent iteration, and since I wanted to be able to solve the problem in a reasonable amount of time, I attempted Q1. Here's the problem: ( APMO 2024/1 ) Let $ABC$ be an acute triangle. Let $D$ be a point on side $AB$ and $E$ be a point on side $AC$ such that lines $BC$ and $DE$ are parallel. Let $X$ be an interior point of $BCED$. Suppose rays $DX$ and $EX$ meet side $BC$ at points $P$ and $Q$, respectively, such that both $P$ and $Q$ lie between $B$ and $C$. Suppose that the circumcircles of triangles $BQX$ and $CPX$ intersect at a point $Y \neq X$. Prove that the points $A, X$, and $Y$ are collinear. Preliminary thoughts First and foremost, here's a diagram: Sorry about th...

(This is Glen.) I was sorting through the LaTeX files on my computer and unearthed an old set of solutions to a mysterious problem set, dated 2020. After a bit of digging in some old Discord servers, I found out that these were solutions to one of Zhao Yu's sets for some RI training, which I had presumably crashed because it was online (thanks to Covid) and I was too free or something. Anyway, this file contained a lengthy introduction to UFDs, which I had recently learnt about in uni and had used to overkill a couple of problems in the set. This is, I think, quite suitable for a blog post, so here we are. The fundamental theorem of arithmetic As a warmup, let's think about something we learn about in primary school (well, at least I remember learning about this in primary school but I am old so this may no longer be the case): the unique prime factorisation of integers. (Fundamental theorem of arithmetic) Each integer $n>1$ can be written uniquely as $n=p_1\cdots p_k$, wher...