Today we shall look at a fairly standard type of argument concerning prime divisors of integer sequences. Given a sequence $latex (a_n)_{n\ge 1}$, we say a prime $latex p$ is a prime divisor of $latex (a_n)$ if $latex p$ is a factor of some term of the sequence. The set of all prime divisors of… Continue reading Prime Supports
Cyclic Digits
Today we shall have a look at a neat little trick that is more commonly seen in computational contests, but occasionally makes an appearance in Olympiad problems as well.Let $latex X=\overline{d_{n-1}d_{n-2}\dots d_0}_b= \sum_{i=0}^{n-1} d_ib^i$ be an integer in base-$latex b$ where $latex 0 \le d_i \le b-1$ for all $latex 0 \le i \le n-1$,… Continue reading Cyclic Digits
Not all is grey
Another IMO has come and gone, leaving us right where we started. Across 31 years of participation, Sri Lanka has cleared a 50% relative team score1 exactly once, while the average team rel. across the last six years is just 32.8%. One might ask why a nation with such a 'proud intellectual tradition' performs so… Continue reading Not all is grey
D-Excircle of the Orthic triangle
Today we shall have a (very brief) look at the $latex D-$excircle of the orthic triangle which appeared recently on the Vietnam Team Selection Test. This configuration can be transposed to the standard excircle configuration by shifting the reference triangle to the orthic triangle, but certain results are interpreted more naturally in an orthic setting.… Continue reading D-Excircle of the Orthic triangle
Farey Sequences
Today we shall look at a class of sequences with a rich and deep history beginning from their earliest appearance in a problem that appeared in the Ladies' Diary, which have numerous applications in number theory and combinatorics, and are lately making an appearance in math Olympiad problems. The Farey sequence of order $latex n$, denoted $latex \mathcal{F}_n$ is… Continue reading Farey Sequences
Four Kissing Circles
Today we shall explore a more classical configuration which has been studied in various settings -- most notably in terms of inversion. We consider $latex \omega_1$,$latex \omega_2$,$latex \omega_3$ and $latex \omega_4$ (with centers $latex O_1$, $latex O_2$, $latex O_3$ and $latex O_4$) to be four circles such that $latex \omega_i$ and $latex \omega_{i+1}$ are externally… Continue reading Four Kissing Circles
Miquel point of a quadrilateral formed by the H-tangent to (BHC)
Today we shall have a look at another orthocentric configuration, that of the Miquel point of the quadrilateral formed by the intersections of the tangent to circle $latex (BHC)$ at $latex H$ with the sides of $latex \triangle ABC$. Many thanks to Yangqin Yang for contributing several proofs and for saving me from crashing out.… Continue reading Miquel point of a quadrilateral formed by the H-tangent to (BHC)
Intersection of a circle centered on a cevian
Today we shall have a look at a generalized configuration, with certain noteworthy special cases. Problems from this configuration seem to not yet have appeared in olympiads, with the only problem I am aware of being due to Ethan Wang. Also, shoutouts to Yanqing Yang for several key ideas and claims. Since most points are… Continue reading Intersection of a circle centered on a cevian
The Bevan Point
Today we shall look at something different from usual, and turn our attention to a famous triangle center, named after Benjamin Bevan -- the circumcenter of the excentral triangle or the Bevan point $latex Be$, which is Kimberling triangle center $latex X_{40}$. A reasonable amount of work (mainly analytical) has been done regarding this point… Continue reading The Bevan Point
Tangents to the Incircle
Today we shall look at a rather interesting configuration which has appeared in only a handful of problems to date with rather surprising and occasionally obscure results. Our work shall explore the tangents from the point $latex T$ which is defined as the intersection of the tangents from $latex B$ and $latex C$ to the… Continue reading Tangents to the Incircle
Angular Momentum 