Olympiad NT Book

New Update in 2025: YouTube Videos to accompany the textbook

Once a week, a new video will be released on this page that supports the textbook. Be sure to subscribe to my YouTube channel so you don’t miss any of the updates! 

Sample Videos

Fibonacci Number Counting #maths #education #fibonacci #mathematics
We introduce the Fibonacci number sequence, a method for counting with the sequence, prove several important identities about the sequence, and finally prove Binet's formula at the end of the video. These examples are taken from Section 1.4 of the textbook "Olympiad Number Theory through Challenging Problems": https://numbertheoryguy.com/publications/olympiad-number-theory-book/
Fibonacci Number Identities
This video answers the question: What is the probability that at least two people in a group of n people share the same birthday? We then use this answer to determine the smallest n for which this probability is at least 1/2. We then explore some further results related to the paradox.
Birthday Paradox
We introduce the principle of strong induction after briefly reviewing the concept of induction. We then use this to solve two challenging problems from the USAMO and the Putnam exam. The examples in this section are drawn from Section 1.2 of the textbook "Olympiad Number Theory Through Challenging Problems": https://numbertheoryguy.com/publications/olympiad-number-theory-book/.
Strong Induction with a Challenging USAMO and Putnam Problem
Sum of Rows of Pascal's Triangle #maths #mathematics #education #numbertheory
We review five binomial identities, including the sum of rows of Pascal's triangle, the sum of alternating rows of Pascal's triangle, and more. We demonstrate how to apply the binomial theorem, along with some algebraic simplifications, to prove some surprising results. The resources are taken from Section 1.3 of my textbook "Olympiad Number Theory through Challenging Problems": https://numbertheoryguy.com/publications/olympiad-number-theory-book/
5 Binomial Identities in 5 Minutes
Intuitive Binomial Theorem Proof #mathematics #maths #mathshorts #algebra #education
We introduce factorials and binomial coefficients, then prove an interesting example with factorials, Pascal's identity, the sum of squares formula, and the Binomial Theorem. These examples are taken from Section 1.3 of the textbook "Olympiad Number Theory through Challenging Problems": https://numbertheoryguy.com/publications/olympiad-number-theory-book/
Proving Pascal's Identity, Binomial Theorem, and Sum of Squares Formula
Sum of Odd Positive Integers with Induction #mathematics #numbertheory #algebra #education #maths
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Olympiad Number Theory Through Challenging Problems

Fourth Edition (pdf)

Fourth Edition, last posted April 21st, 2025, first two chapters added.

Third Edition (pdf) 

Third Edition published December 18th, 2016.

An 120 page introductory Olympiad Number Theory book I wrote for anyone with a passion for number theory and problem solving published on the Art of Problem Solving website. Each section begins by introducing a main concept or idea, and then contains many engaging and challenging problems. My goal for the text was to show how several problem solving skills–experimenting with small cases, simplification, and pattern finding–can be used to tackle a wide variety of Olympiad Number Theory problems. There are five chapters on Divisibility, Modular Arithmetic, p-adic valuation, Diophantine Equations, and Problem Solving Strategies.

Second Edition

(3/2/2015)  Olympiad Number Theory Through Challenging Problems (pdf) 

First Edition

(8/25/2013) The Basics of Olympiad Number Theory (pdf) (Problems and Theorem)