SIMO Retiree Blog

Glen here. At some point about a year ago, I scrolled through the blogger interface and found an old draft by Dylan, a skeletal outline of a post featuring a problem and a few section headings. The problem happened to be one I recognised: Show that $2^{2000}$ divides the numerator of $2+\frac{2^2}2 + \frac{2^3}3 + \cdots + \frac{2^{2024}}{2024}$ when expressed as a simplified improper fraction. I remembered first seeing this as a bonus problem in some old SIMO National Team set, when I'd looked at it, had absolutely no idea how to approach it, and promptly given up. But this time, having learnt some more math in the intervening years, I did  have an idea of what to do, and ended up solving it it in my head during some seminar that I wasn't quite following. The solution turned out to be pretty cool, so I made a note to myself to write a blog post about it at some point if Dylan's post never manifested. I then promptly forgot about it. Fast forward to this week. Some of the i...

(Guest author Andrew here!) A good exercise for Olympiad students is to prove the following: A real polynomial $f(x)\in \R[x]$ which is everywhere non-negative ( psd , short for positive semi-definite) must be a sum of squares ( sos ).  This is in some sense an algebraic witness to the analytic property of being non-negative, and is an example of what is known as a Positivstellensatz . One might reasonably ask whether this extends to the case of multiple variables, and it turns out the answer is no .  Consider the two variable polynomial $g(x,y)=x^4y^2+x^2y^4-3x^2y^2+1$ (the Motzkin polynomial ). Non-negativity follows from the AM-GM inequality. But how can we show that it isn't a sum of squares? We now introduce an object known as the Newton polytope. Given a polynomial $f$,  To every monomial with non-zero coefficient, assign a point whose $i^{th}$ coordinate is the degree of the $i^{th}$ variable in that monomial, e.g. assign to $x_1^n$ the point $(n, 0)$. The Newton p...

(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...