Hi everyone!

It's been quite some time since I wrote two previous articles in the cycle:

Part 1: Introduction
Part 2: Properties and interpretation
Part 3: In competitive programming

This time I finally decided to publish something on how one can actually use continued fractions in competitive programming problems.

Few months ago, I joined CP-Algorithms as a collaborator. The website also underwent a major design update recently, so I decided it would be great to use this opportunity and publish my new article there, so here it is:

It took me quite a while to write and I made sure to not only describe common competitive programming challenges related to continued fractions, but also to describe the whole concept from scratch. That being said, article is supposed to be self-contained.

Beyond what I already wrote on continued fractions in my previous blog posts, I added several explanatory examples and pictures. I also elaborated on the connection between continued fractions and the Stern-Brocot tree and Calkin-Wilf tree, binary trees that contain all distinct positive rational numbers. Also, I really hope that I managed to simplify the general story-telling in the article.

To make a further teaser, here are the major problems that are dealt with in the article:

• Given $$$r$$$ and $$$m$$$, find the minimum value of $$$q r \pmod m$$$ on $$$1 \leq q \leq n$$$.
• Given $$$p$$$, $$$q$$$ and $$$b$$$, construct the convex hull of lattice points below the line $$$y = \frac{px+b}{q}$$$ on $$$0 \leq x \leq n$$$.
• Given $$$A$$$, $$$B$$$ and $$$C$$$, find the maximum value of $$$Ax+By$$$ on $$$x, y \geq 0$$$ and $$$Ax + By \leq C$$$ (Timus — Crime and Punishment).
• Given $$$p$$$, $$$q$$$ and $$$b$$$, compute the following sum (Library Checker — Sum of Floor of Linear, June Challenge 2017 — Euler Sum):

• Given $$$r$$$ and $$$m$$$, find $$$\frac{p}{q}$$$ such that $$$p, q \leq \sqrt{m}$$$ and $$$p q^{-1} \equiv r \pmod m$$$ (102354I - From Modular to Rational).
• Given $$$\frac{0}{1} \leq \frac{p_0}{q_0} < \frac{p_1}{q_1} \leq \frac{1}{0}$$$, find $$$\frac{p}{q}$$$ such that $$$(q,p)$$$ is lexicographically smallest and $$$\frac{p_0}{q_0} < \frac{p}{q} < \frac{p_1}{q_1}$$$ (GCJ 2019, Round 2 — C).

There are likely much more problems where continued fractions are used, please mention them in the comments if you know any!

Finally, since CP-Algorithms is supposed to be a wiki-like project (that is, to grow and get better as time goes by), please feel free to comment on any issues that you might find while reading the article, ask questions or suggest any improvements. You can do so in the comments below or in the issues section of the CP-Algorithms GitHub repo. You can also suggest changes via pull request functionality.