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2017 | Book

The Colorado Mathematical Olympiad: The Third Decade and Further Explorations

From the Mountains of Colorado to the Peaks of Mathematics

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About this book

Now in its third decade, the Colorado Mathematical Olympiad (CMO), founded by the author, has become an annual state-wide competition, hosting many hundreds of middle and high school contestants each year. This book presents a year-by-year history of the CMO from 2004–2013 with all the problems from the competitions and their solutions. Additionally, the book includes 10 further explorations, bridges from solved Olympiad problems to ‘real’ mathematics, bringing young readers to the forefront of various fields of mathematics. This book contains more than just problems, solutions, and event statistics — it tells a compelling story involving the lives of those who have been part of the Olympiad, their reminiscences of the past and successes of the present.

I am almost speechless facing the ingenuity and inventiveness demonstrated in the problems proposed in the third decade of these Olympics. However, equally impressive is the drive and persistence of the originator and living soul of them. It is hard for me to imagine the enthusiasm and commitment needed to work singlehandedly on such an endeavor over several decades.

<—Branko Grünbaum, University of Washington

After decades of hunting for Olympiad problems, and struggling to create Olympiad problems, he has become an extraordinary connoisseur and creator of Olympiad problems. The Olympiad problems were very good, from the beginning, but in the third decade the problems have become extraordinarily good. Every brace of 5 problems is a work of art. The harder individual problems range in quality from brilliant to work-of-genius… The same goes for the “Further Explorations” part of the book. Great mathematics and mathematical questions are immersed in a sauce of fascinating anecdote and reminiscence. If you could have only one book to enjoy while stranded on a desert island, this would be a good choice.

<—Peter D. Johnson, Jr., Auburn University

Like Gauss, Alexander Soifer would not hesitate to inject Eureka! at the right moment. Like van der Waerden, he can transform a dispassionate exercise in logic into a compelling account of sudden insights and ultimate triumph.

— Cecil Rousseau Chair, USA Mathematical Olympiad Committee

A delightful feature of the book is that in the second part more related problems are discussed. Some of them are still unsolved.

—Paul Erdős

The book is a gold mine of brilliant reasoning with special emphasis on the power and beauty of coloring proofs. Strongly recommended to both serious and recreational mathematicians on all levels of expertise.

—Martin Gardner

Table of Contents

Frontmatter

The Third Decade

Frontmatter
Twenty-First Colorado Mathematical Olympiad
April 16, 2004
Abstract
Mark Heim of Loveland Wins Again
Alexander Soifer
Twenty-Second Colorado Mathematical Olympiad
April 22, 2005
Abstract
Mark Heim of Loveland Wins for the Third Straight Time!
Alexander Soifer
Twenty-Third Colorado Mathematical Olympiad
April 21, 2006
Abstract
Sam Elder of Fort Collins solves “The Famous Five”
Alexander Soifer
Twenty-Fourth Colorado Mathematical Olympiad
April 20, 2007
Abstract
Sam Elder and Hannah Alpert Shine Again
Alexander Soifer
Twenty-Fifth Colorado Mathematical Olympiad
April 18, 2008
Abstract
Colorado Mathematical Olympiad Celebrates a Quarter a Century!
Alexander Soifer
Twenty-Sixth Colorado Mathematical Olympiad
April 17, 2009
Abstract
Colorado Springs sophomore Alan Gardner wins gold
Alexander Soifer
Twenty-Seventh Colorado Mathematical Olympiad
April 23, 2010
Abstract
Colorado Springs Allan Gardne r wins again!
Alexander Soifer
Twenty-Eighth Colorado Mathematical Olympiad
April 22, 2011
Abstract
Three New Olympians Win, while the Second Olympiad Book is born
Alexander Soifer
Twenty-Ninth Colorado Mathematical Olympiad
April 20, 2012
Abstract
And the Winner is Albert Soh of Boulder
Alexander Soifer
Thirtieth Colorado Mathematical Olympiad
April 26, 2013
Abstract
Celebrating the Centenary of Paul Erdős ’ Birth
Alexander Soifer
A Round Table Discussion of the Olympiad, or Looking Back from a 30-Year Perspective
Abstract
It is impossible to translate into a written word the excitement of the 30-Year Anniversary Award Presentation and the Round Table Panel that took place. Yet, let me attempt to give you a glimpse of this event.
Alexander Soifer

Further Explorations of the Third Decade

Frontmatter
Introduction to Part II
Abstract
What does the Aristotle mean by this paradoxically sounding statement: “The more you know, the more you know you don’t know? Of course, the more you know, the more you know! However, the more you know, the better you understand how much more there is to know! It is like climbing a mountain: the higher your rise, the wider horizon you see.
Alexander Soifer
E21. Cover-Up with John Conway, Mitya Karabash, and Ron Graham
Abstract
The problem title comes from a famous proverb “To have a cake and eat it too,” which in my early American years I could not understand. Surely you have to “have a cake” in order “to eat it”! A better formulation of this folk wisdom would have been “To keep a cake and eat it too,” which is obviously impossible, hence a moral of the proverb. But that is not what I would like to share with you here. A version of what follows first appeared in the second, 2009 Springer edition of my book “How Does One Cut a Triangle?” [Soi6]. But it fits here so well that I am including its new updated and expanded 2016 version.
Alexander Soifer
E22. Deep Roots of Uniqueness
Abstract
Obviously, any solution of problem 21.5.(B) can be presented in a form of 21 checkers on a 7 × 7 board (see left 7 × 7 part with 21 black checkers in Fig. 18). It is not obvious that the solution is unique, i.e., by a series of interchanges of rows and columns, any solution of this problem can be brought to match precisely the one I presented in Fig. 18! Of course, such interchanges mean merely renumbering of players of the same teams.
Alexander Soifer
E23: More about Love and Death
Abstract
I hope you did not take the DNA’s featured in my problem 22.5 to faithfully reflect reality. Remember, we are in the Illusory World of Mathematics! To whet your appetite for the problem, I invented the bacterium bacillus anthracis, causing anthrax (death), in problem 22.5.(A). In problem 22.5.(B), I went even further by imagining the bacterium bacillus amoris, causing love. I was inspired by a talk by a Ph.D. student Martin Klazar that I attended in 1996 during my long term visit of Charles University in the beautiful Prague, Czech Republic. Now Martin is a professor at that same university.
Alexander Soifer
E24: One Amazing Problem and Its Connections to Everything—A Conversation in Three Movements
Abstract
Inspired by problem 24.5: “Natural Split”
Alexander Soifer
E25: The Story of One Old Erdős Problem
Abstract
Inspired by problem 25.5: “One Old Paul Erdős’ Problem”
Alexander Soifer
E26: Mark Heim’s Proof
Abstract
Inspired by problem 22.4(B): “Red and White”
Alexander Soifer
E27: Coloring Integers—Entertainment of Mathematical Kind
Abstract
Inspired by problem 27.5: “Colorful Integers”
Alexander Soifer
E28: The Erdős Number and Hamiltonian Mysteries
Abstract
Inspired by problem 23.5 “Math Party”
Alexander Soifer
E29: One Old Erdős–Turán Problem
Abstract
Inspired by problem 30.5: “One Old Erdős Problem”
Alexander Soifer
E30: Birth of a Problem: The Story of Creation in Seven Stages
Abstract
Problem solving requires the existence of problems. Somebody has to create them. But how?
Alexander Soifer

Olympic Reminiscences in Four Movements

Frontmatter
Movement 1. The Colorado Mathematical Olympiad Is Mathematics; It Is Sport; It Is Art. And It Is Also Community, by Matthew Kahle
Abstract
Professor Soifer asked me to share some reminiscences for his new book. I am happy to! First, I will reprint parts of an essay I wrote in 2008, for an earlier Olympiad book.
Alexander Soifer
Movement 2. I’ve Begun Paying Off My Debt with New Kids, by Aaron Parsons
Abstract
Permit me first reproduce a section about Aaron Parsons from my 2011 Olympiad book [Soi9].
Alexander Soifer
Movement 3: Aesthetic of Personal Mastery, by Hannah Alpert
Abstract
Hannah Alpert was the first girl to win back to back second prizes in the Colorado Mathematical Olympiad in 2006 and 2007. Moreover, she was a solo second prize winner behind only Sam Elders, who was a solo first prize winner. Before that, as a sophomore in 2005, Hannah won third prize, and was, together with the winner Mark Heim, the only Olympian to solve problem 22.4, in her own way, unknown to me.
Alexander Soifer
Movement 4. Colorado Mathematical Olympiad: Reminiscences by Robert Ewell
Abstract
Robert N. (Bob) Ewell is a graduate of Clemson University (B.S. in Mathematics, 1968). He did graduate study in mathematics at Clemson and the University of Nebraska. He went on to pick up a master’s in Education from Troy State University in 1982, and a Doctor of Education from Auburn University in 1984. An AFROTC cadet at Clemson University, he served in the US Air Force for 20 years, retiring as a lieutenant colonel in 1990. After the Air Force, he did statistical consulting for 10 years before entering Christian ministry. He now serves with The Navigators as a leadership development coach. In addition to being a fine amateur mathematician, Bob plays piano and enjoys playing and watching sports. Here is Bob’s story.
Alexander Soifer
Farewell to the Reader
Abstract
Does Dostoyevsky refer to mathematics in the epigraphs above? Probably not, but his lines are so applicable to mathematics, with their references to proving existence and importance of beauty! Einstein, a connoisseur of Beauty, observes poetic qualities of pure mathematics. Mathematics could be viewed as a science whose truths exist in Nature independently of our mind. It is also used as a tool in a variety of other sciences. In a sense, mathematics is a language, used by many disciplines to make themselves more rigorous. And mathematics could be viewed as an art, which not only reflects Nature, but also creates Beauty that can compete with the Beauty of Nature. The Colorado Mathematical Olympiad exists to spread this Beauty, and to pass this connoisseurship to young minds.
Alexander Soifer
Backmatter
Metadata
Title
The Colorado Mathematical Olympiad: The Third Decade and Further Explorations
Author
Alexander Soifer
Copyright Year
2017
Electronic ISBN
978-3-319-52861-8
Print ISBN
978-3-319-52859-5
DOI
https://doi.org/10.1007/978-3-319-52861-8

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