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 $$$Ax + By \leq C$$$.
• Given $$$p$$$, $$$q$$$ and $$$b$$$, compute the following sum:

• Given $$$r$$$ and $$$m$$$, find $$$\frac{p}{q}$$$ such that $$$p, q \leq \sqrt{m}$$$ and $$$p q^{-1} \equiv r \pmod m$$$.
• 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 $$$\frac{p_0}{q_0} < \frac{p}{q} < \frac{p_1}{q_1}$$$.

So far, here is the list of problems that are explained in the article:

• Timus — Crime and Punishment
• June Challenge 2017 — Euler Sum
• NAIPC 2019 — It's a Mod, Mod, Mod, Mod World
• Library Checker — Sum of Floor of Linear
• 102354I - From Modular to Rational
• GCJ 2019, Round 2 — New Elements: Part 2

And an additional list of practice problems where continued fractions could be useful:

• UVa OJ — Continued Fractions
• Codeforces Round #184 (Div. 2) — Continued Fractions
• Codeforces Round #325 (Div. 1) — Alice, Bob, Oranges and Apples
• POJ Founder Monthly Contest 2008.03.16 — A Modular Arithmetic Challenge
• 2019 Multi-University Training Contest 5 — fraction
• SnackDown 2019 Elimination Round — Election Bait
• Tavrida NU Akai Contest — Continued Fraction

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.