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
Tag: olympiad
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
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
Discrete Continuity in Combinatorics
This blog post corresponds to my newest olympiad handout on discrete continuity. The concept of discrete continuity is an extremely effective and versatile method in tackling a wide variety of combinatorics problems. This technique has been used extensively for many years especially in particular classes of problems. Here, after a quick recap of the basics we shall… Continue reading Discrete Continuity in Combinatorics
Bulgarian Solitaire
In this article we provide investigate a famous classical process involving piles of stones (or originally cards) introduced by Martin Gardner. The game involves a set of piles of stones with $latex \frac{n(n+1)}{2}$ stones in total. In each move, the topmost stone in each pile is picked and a new pile is formed with these… Continue reading Bulgarian Solitaire
Miquel Point of the Angle Bisectors
This article will explore a relatively new configuration which I have talked about extensively in my handout Neat Config which was crafted from a series of notes made by Om Kutte for lectures conducted by him at the (Unofficial) Indian National Mathematical Olympiad Training Camp. This is the master diagram. All the point definitions (and… Continue reading Miquel Point of the Angle Bisectors
Some Length Conditions on a Triangle
In this post we will have a look at three really common triangle length conditions. They look almost the same but their slightly different definitions invoke highly contrasting configurations. After introducing each condition, and looking at how it can be interpreted, we solve (at least partially) some contest problems involving those conditions. 1. Equal Lengths… Continue reading Some Length Conditions on a Triangle
Angular Momentum 