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## Über dieses Buch

Over the past two decades, the once small local Colorado Springs Mathematics Olympiad, founded by the author himself, has now become an annual state-wide competition, hosting over one-thousand high school contenders each year.

This updated printing of the first edition of Colorado Mathematical Olympiad: the First Twenty Years and Further Explorations offers an interesting history of the competition as well as an outline of all the problems and solutions that have been a part of the contest over the years. Many of the essay problems were inspired by Russian mathematical folklore and written to suit the young audience; for example, the 1989 Sugar problem was written as a pleasant Lewis Carroll-like story. Some other entertaining problems involve old Victorian map colorings, King Arthur and the knights of the round table, rooks in space, Santa Claus and his elves painting planes, football for 23, and even the Colorado Springs subway system.

The book is more than just problems, their solutions, and event statistics; it tells a compelling story involving the lives of those who have been part of the Olympiad from every perspective.

## Inhaltsverzeichnis

### Colorado Mathematical Olympiad: How it Started and What it has Become

The Colorado Mathematical Olympiad. Why did I create it? Who needs it? Who makes it possible? To answer these questions, I have to start with my own story and a critical role mathematical Olympiads played in it.

Alexander Soifer

### Three Celebrated Ideas

In the Colorado Mathematical Olympiad the same problems are offered to every participant from a seventh grader to a senior. This is why they must require minimal bits of knowledge for their solutions, such as the sum of angles in a triangle is equal to 180°, or the three bisectors of a triangle have a point in common. These problems do require a great deal of common sense, creativity, and imagination. Some of the problems model mathematical research: they would capitulate only to experimenting with particular cases, followed by noticing a pattern, followed in turn by generalization, formulation of a hypothesis, and finally by a proof.

Alexander Soifer

On April 27, 1984, the few (170 to be exact), the curious, and the brave showed up at the University of Colorado at Colorado Springs (UCCS). They did not know what to expect. Neither did the journalists from the newspapers and television stations. Here is how Julie Bird described the event the next morning on the front page of the Metro Section of

The Gazette Telegraph

.

Alexander Soifer

The First Olympiad was such a success, it received such wonderful media coverage, that I was called into the office of my dean. He went straight to the point: “You ran the Olympiad, I solicited the prizes. What do we need the College of Education for?” “You want to join Education as a co-organizer?” I asked. “No, Engineering alone will run the Olympiad from now on.” “I cannot go to Dennis (Dean of Education), the only person who supported the Olympiad from the start, and tell him to get out.” “I’ll do it myself,” said my dean.

Alexander Soifer

Alexander Soifer

In early fall 1986 I received a surprising call from Ms. Bailey Barash, an anchorwoman at CNN, whom I saw daily on my TV screen. A CNN news team was to come to Colorado Springs and cover an important international gathering at the Broadmoor Hotel. A CNN education crew wanted to come along and cover the Olympiad, if it were to be “served” in November. I did not feel we could make such a move, even for CNN. On CNN’s request, I reminded them in March about their interest in covering the Olympiad, but, of course, without a major political event in the neighborhood they

were

interested in the Olympiad, but not

too

interested.

Alexander Soifer

As you recall from Historical Notes 3, Dean Tracey and my Chair, Rangaswami, ordered me to close down the Olympiad, and threatened with consequences to my employment if I did not. In 1987 Rangaswami, using the draft by another perpetrator in my tenure process, Professor Gene Abrams, produced “Proposal to the CCHE (The Colorado Commission on Higher Education) for the Promotion and Encouragement of Excellence.” Rangaswami asked for funding of the “Center of Resources for Excellence in and Advancement of Mathematics (CREAM).”

Alexander Soifer

On June 23, 1988, UCCS Chancellor Dwayne Nuzum offered to the Olympiad a tough, yet prestigious test. He asked me to make a presentation about the Olympiad to the Board of Regents of the University of Colorado (four-campus system). I told the regents what the Olympiad was, how the entire community (education, industry, and government) joined together in this ongoing project and why I founded it. The Olympiad passed the test.

Alexander Soifer

In August 1989 I taught at the International Summer Institute in Long Island, New York. A fine international contingent of gifted high school students for the first time included a group from the Soviet Union. Some members of this group turned out to be mathematics Olympiad “professionals,” winners of the Soviet Union National Mathematics and Physics Olympiads. There was nothing in the Olympiad genre that they did not know. There was no point in teaching these kids problem solving. Instead I offered them – and everyone else – an introduction to certain areas of combinatorial geometry. We quickly reached the forefront of mathematics, full of open problems.

Alexander Soifer

The eighth Olympiad attracted the largest number of contestants in the history of the event: 987. As in the previous year, a much greater number of junior high school students participated this time. A good number of junior high school students won awards, especially from the classes of two wonderful teachers: John Putnam and my former student Lossie Ortiz.

Alexander Soifer

This year we featured a truly international set of problems. Problem 9.4 was contributed by the legendary Paul Erd?os of the Hungarian Academy of Sciences. Problem 9.3 was inspired by a problem my friend and the leading Abelian group theorist Laszlo Fuchs of Tulane University mentioned in passing as we strolled down his street in New Orleans (Laszlo was born in Hungary). Problem 9.5 was created by Dr. Pak-Hong Cheung of Hong Kong (who created the first inequality in Problem 9.5(A)) and me (the second inequality and Problem 9.5(B)). The two easy Problems 9.1 and 9.2 were suggested by me.

Alexander Soifer

Alexander Soifer

### Introduction to Part II

When a problem is solved, seldom does a mathematician ask what is next. With his insightful mind the great playwright Bernard Shaw observes that a problem is never solved without giving birth to a number of new problems. My book

How Does One Cut a Triangle?

[S2], [S13] is about this process. It traces the process of going from one problem to another, then another, etc., all the way to the forefront of mathematics, to open problems. In this book we will briefly look at these two trains of mathematical thought (see Chapters E3 and E4 of

Further Explorations

) and several others.

Alexander Soifer

### E1. Rooks In Space

Problem 1.5 was another form of the following problem created by Semjon Slobodnik and myself in 1972 and published in 1973, [SS].

Alexander Soifer

### E2. Chromatic Number of the Plane (My Favorite Open Problem)

This

is

my favorite open problem in all of mathematics, so much so that by the time this new Springer edition is coming out, I have written

The Mathematical Coloring Book

[S11] about this and related problems. In the first 1994 edition of this book I remarked that

The Coloring Book

will appear in 1995, and the following essay is essentially an excerpt from its Chapter II. In fact,

The Mathematical Coloring Book

came out froom Springer much later, on November 4, 2008, the day we elected President Barack Obama. I gave it 18 years and all I had in terms of my knowledge of the arts, sciences, poetry, philosophy, aesthetics, and in terms of all my energy and passion. I hope you will read and enjoy that 640-page magnum opus. I will leave this essay more or less as it originally appeared in 1994, with a new mild editing – it will now serve as an introduction to

The Coloring Book

.

Alexander Soifer

### E3. Polygons in a Colored Circle, Polyhedra in a Colored Sphere

With the method used in the solution of Problem 3.3 we can solve other problems as well.

Alexander Soifer

### E4. How Does One Cut a Triangle?

The solution of problem 4.4 showed that not every triangle can be cut into two triangles similar to each other, and on the other hand, every triangle can be cut into six triangles similar to each other. In 1970, still an undergraduate student, I posed and solved the following two much more general problems.

Alexander Soifer

### E5. Points in Convex Figures

During the summer of 1987, I taught at the International Summer Institute in Orange, California. Students came from the U.S., Japan, Israel, Hungary, Switzerland, and France. For their test I decided to create a problem requiring the use of the Pigeonhole Principle in geometry. I came up with Problem 5.4(A).When you solve a problem like that, you ask yourself, can I prove a stronger result, i.e., a result with a smaller

n

? This train of thought led me to problem 5.4(B), and consequently to problem 5.5(A). The problem became too good to be used for a test. I saved it for the Fifth Colorado Mathematical Olympiad. Problem 5.5(B) shows that the result of problem 5.5(A) is best possible: you cannot reduce

n

to below 5. Does it mean that we have reached the end of the road? Not at all! Instead of looking at triangles alone, we can include all

convex

figures.

Alexander Soifer

### E6. Triangles in a Colored Plane

Problem 7.5(A) and 7.5(B) delivered us examples of triangles that exist

monochromatically

in the plane, no matter how we color it in 2 colors. A triangle in a colored plane is called

monochromatic

if all of its vertices are colored in the same color.

Alexander Soifer

### E7. Rectangles in a Colored Plane

The result of problem 8.5(A) was improved in problem 8.5(B). Can we strengthen it further? How much further? It is an open problem. In [S5] I offered a \$25 prize for the first solution of this problem. It is still unclaimed in 2010.

Alexander Soifer

### E8. Colored Polygons

I would like to go back to April of 1970 one more time (Chapter E4 included some reminiscences of that time). The thirty judges of the Fourth Soviet Union National Mathematical Olympiad, of whom I was one, stayed at a fabulous white castle, halfway between the cities of Simferopol and Alushta in sunny Crimea, surrounded by the Black Sea. The problems had been selected and approved in a meeting of all thirty judges and A. N. Kolmogorov. They were being printed. The Olympiad was to take place the next morning, when something shocking occurred.

Alexander Soifer

### E9. Infinite-Finite

We looked at two very different solutions of Problem 10.4(A). The first solution was quite powerful.

Alexander Soifer

### E10. The Schur Theorem

As you probably know, in the 1630’s, the genius mathematician Pierre de Fermat formulated his so-called Last Theorem in the margin of Diaphantus’s

Arithmetica

.

Alexander Soifer

### The Olympiad: How It Has Continued and What It Has Become

has survived another decade (actually by now it has survived for over a quarter a century). Looking back, I can attest that it has become more sophisticated, more closely linked to research mathematics. At the request of the Executive Director of mathematics at Springer, Ann Kostant, I am adding the second decade of the Olympiad’s history and problems in this new Springer edition. However, I cannot just add the Olympiad material without adding its links to and shared essence with “real” mathematics. I therefore am adding 10 new

Further Explorations

to emphasize – and celebrate – the togetherness of the Olympiads and Mathematics. These, now 20, Explorations are the bridges from the Olympiads to mathematical research. Walk across them – and you find yourself in the magical world of mathematics!

Alexander Soifer

The Olympiad season began on April 18, 1994, when the

Gazette Telegraph

published the article “Olympiad Showcases Math’s Beauty, Elegance” by Teresa Owen-Cooper.

Alexander Soifer

A heavy snow was unable to stop six hundred and three students from coming to the University of Colorado at Colorado Springs for the Twelfth annual Colorado Mathematical Olympiad. Junior high and high school students came from all over the State: Denver, Parker, Fountain, Ca˜non City, Woodland Park, Florissant, Divide, Englewood, Littleton, Castle Rock, Manitou Springs, Falcon, Peyton, Fort Collins, Hayden, Longmont, Aurora, Franktown, Cascade, Brush, Merino, Hillrose, Ordway, Olney Springs, Erie, Lafayette, Calhan, Yoder, Rush, Sedalia, Larkspur, Monument, Widefield, and Colorado Springs. For the first time in twelve years we also had participants who came from elementary schools.

Alexander Soifer

The thirteenth Annual Colorado Mathematical Olympiad brought together some 600 junior high and high school students. Contestants came from all over Colorado: Denver, Parker, Canon City, Woodland Park, Littleton, Castle Rock, Fort Collins, Manitou Springs, Castle Rock, Rangely, Ellicott, Durango, Longmont, Foxfield, Erie, Franktown, Sanford, Falcon, Calhan, Fountain, Aurora and Colorado Springs.

Alexander Soifer

The Olympiad season commenced with the April 17, 1998,

Gazette

article “Calling all creative types to state Math Olympiad” by Wendy Y. Lawton: Number geeks need to apply. Calculus fiends, geometry nerds, algebra wonks – forget it.

Alexander Soifer

Unlike a year ago, when the Colorado Mathematical Olympiad was buried by two feet of snow and had to be postponed, this year’s event featured Colorado-blue skies and temperatures in the mid 70s. On April 24, 1998, the Fifteenth annual Colorado Mathematical Olympiad brought together some 500 middle and high school students. Contestants came from all over Colorado: Rangely, Durango, Denver, Thornton, Baker, Commerce City, Aurora, Lakewood, Littleton, Castle Rock, Cascade, Sedalia, Parker, Fort Collins, Manitou Springs, Henderson, Lafayette, Longmont, Erie, Parker, Franktown, Falcon, Peyton, Calhan, Yoder, Ellicott, Simla, Cañon City, Fountain, Monument, Elbert, Black Forest, and Colorado Springs.

Alexander Soifer

Tuesday April 20, 1999, was the gravest day in the history of Colorado. The tragic shooting at Columbine High School near Denver left many students and a teacher dead, others wounded. It shocked every one of us. I was torn by a moral dilemma: should I cancel the Colorado Mathematical Olympiad, scheduled for the coming Friday? Columbine often participated. Last year Columbine’s junior John Batchelder won first prize.Was he OK? If not, I felt, then cancellation was a must. If he were OK, what would he prefer, a cancellation in solidarity with Columbine’s tragedy or holding the Olympiad as a symbol that we were down but not out?

Alexander Soifer

Clouded by the first anniversary of Columbine’s tragedy, the seventeenth annual Colorado Mathematical Olympiad took place on April 21, 2000, under majestically blue Colorado skies and summer temperatures in the 70s. The last in the XX century Olympiad brought together 480 middle and high school students. Contestants came from all over Colorado: Rangely, Florissant, Commerce City, Aurora, Littleton, Adam City, Parker, Fort Collins, Manitou Springs, Henderson, Erie, Parker, Franktown, Peyton, Calhan, Hayden, Kiowa, Ellicott, Cañnon City, Widefield, Monument, Elbert, Elizabeth, Larkspur, and Colorado Springs.

Alexander Soifer

The Olympiad season commenced on April 19, 2001, with Raquel Rutledge’s article “Let the math games begin” in the

Gazette

: Solving one of Alexander Soifer’s math problems is like finding a treasure.

Alexander Soifer

The nineteenth Colorado Mathematical Olympiad (CMO-2002) took place on April 19, 2002, the day before the third anniversary of the Columbine tragedy. It brought together 614 middle and high school students (a 28% increase from the previous year). Contestants came from all over Colorado: Agate, Aurora, Benett, Black Forest, Brighton, Canon City, Cascade, Centennial, Colorado Springs, Deer Trail, Denver, Dinosaur, Divide, Englewood, Falcon, Fort Collins, Fort Lupton, Fort Morgan, Fountain, Fowler, Las Animas, Littleton, Luatkins, Manitou Springs, Monument, Pueblo, Rangely, Weldona, Widefield, andWoodland Park. The organizers were unable to explain how kids from Bagg, Wyoming, slipped into the competition :-).

Alexander Soifer

In December 2002 I left my home in Colorado Springs to serve as a Long Term Visiting Scholar at The Center for Discrete Mathematics and Theoretical Computer Science (DIMACS), a joint project of Princeton University, Rutgers University, and a few top research industrial labs), and as a Visiting Fellow at Princeton University’s Mathematics Department. It was a privilege to work in the historic Fine Hall of Princeton-Math, and to befriend senior legendary colleagues, such as Harold W. Kuhn (who nominated the “Beautiful Mind” John Nash for the 1994 Nobel Prize), Edward Nelson (creator of the chromatic number of the plane problem), John H. Conway, Yakov and Elena Sinai, Robert Gunning, the great historian of science Charles Coulston Gillispie, pre-Columbian and ancient art expert and collector Gillette Griffin (in his house, I got to hold inmy hands a torso byMichelangelo and drink wine out of fifth century BC ancient Greek cups). I gave talks at the Institute for Advanced Study, Princeton-Math, Rutgers-Math, Claude Shannon Laboratory of AT&T.

Alexander Soifer

### How I Envision Mathematics Education: Introduction to Part IV

When I was in middle school – in grades 6, 7, 8 – my first Olympiads showed that there is mathematics unlike the one in school, beautiful, humorous, surprising.

Alexander Soifer

### E11. Chromatic Number of a Grid

As you recall, Problem 11.4 was created by Professor John Horton Conway of Princeton University and me on March 8, 1994, when we met at the “International Southeastern Conference on Combinatorics, Graph Theory, and Computing” at Florida Atlantic University in Boca Raton, Florida. I had heard a lot about John, and read his (jointly with Elwyn Berlecamp and Richard Guy) spectacular book

Winning Ways for Your Mathematical Plays

. John’s talk about quantum mechanics was very entertaining (at one point he even jumped up in the air :-) and inspiring.

Alexander Soifer

### E12. Stone Age Entertainment

In Problem 14.1, the game allowed a nice solution for the pile of 99 pebbles.

Alexander Soifer

### E13. The Erdős Train Station

Paul Erdős’s articles remind me of train stations. They do not merely report a result or two that the author proved. Each Erdős’s classic papers is a treasure trove of many trains of thought. Let us take a ride here on a couple of such trains from the Erd?os Station-1946 [E0].

Alexander Soifer

### E14. Squares in Square: The 1932 Erdős Conjecture

I will take you here on a journey through problems, conjectures, and results. My goal is to present a live fragment of mathematics and to show how mathematical train of thought travels. I hope you will enjoy the ride!

Alexander Soifer

### E15. From Squares in a Square to Clones in Convex Figures

As I thought about Paul Erdős’s problem, it appeared natural for me to pose a “dual” problem, and thus give birth to the

New Squares in a Square Problem.

Alexander Soifer

### E16. Olde Victorian Map Colouring

Let us start with the history of the arguably second most famous problem in the entire history of mathematics. It commences in Victorian London in the year 1852, when the 20-year old Francis Guthrie created The Four-Color Conjecture (4CC), and continues for 124 years, when in 1976 Kenneth Appel and Wolfgang Haken, with the assistance of John Koch and over 1200 hour of mainframe computer time converted 4CC into 4CT – The Four Color Theorem. A second proof had to wait another 20 years: in 1997 Neil Robertson, Daniel Sanders, Paul Seymour and Robin Thomas also proved 4CT. Both proofs required an essential use of computing.

Alexander Soifer

### E17. Achievement Games: Is Snaky a Loser?

Problems 17.3(A) and 17.3(B) come from “achievement” games invented apparently in 1976 by the famous graph theorist Frank Harary, author of over 700 papers and the well known (I enjoyed reading it in Russian translation) 1969 book

Graph Theory

. Frank wrote an article [Har] about these games for

Geombinatorics

that was “Dedicated toMartin Gardner, who gave Snaky its name.” Therefore I have a rare opportunity and distinct pleasure of giving the microphone to the inventor.

Alexander Soifer

### E18. Finite Sets of Infinite Fun: Three Classic Problems

Did you enjoyed the Olympiad Problem 19.4? Then brace yourself for much more fun! While all explorations in different ways are dear to my heart, this small collection may provide the greatest enjoyment of all. Some of these problems have been solved, others are still await- ing their conqueror, but all of them satisfy the definition of “classic” problems of mathematics.

Alexander Soifer

### E19. From King Arthur to Mysteries of Hamiltonian Graphs

As you will recall, in the solution of Problem 19.5 we introduced for our convenience the notion of

Hamiltonian cycle

, or a walk along the edges through all vertices of a graph, that does not repeat a vertex (and therefore, does not repeat any edge) until it ends at the vertex it started from. When such a Hamiltonian cycle exists, the graph is called

Hamiltonian.

Alexander Soifer

### E20. Exploring Map Coloring

The point of this problem is to “contradict” the Four-Color Theorem. In fact, problem 20.5 illuminates this famous theorem in a novel way.

Alexander Soifer

### Introduction to Part V

While preparing this new much-expanded edition, a thought visited me: Let me offer a microphone (or is it a writing feather or a com- puter keyboard?) to the past winners, to share what they thought about the Colorado Mathematical Olympiad, how it affected their lives, and what their lives AD (

After Departure

from high school and the Olympiad) have been like.

Alexander Soifer

### Reminiscences in Eight Parts

Rick Ansorge,

Gazette Telegraph

, Thursday, April 22, 1993: Math problems that make most people’s eyes glaze over bring a sparkle to the eyes of Alexander Soifer. Soifer – a math prodigy, Soviet 'emigr'e and professor at the University of Colorado at Colorado Springs – is the driving force behind the Colorado Mathematical Olympiad. The 10th–annual event, which begins at 8:15 a.m. Friday in UCCS’s Science Hall, will pit the minds of up to 1,100 junior and senior high school students against five mad– deningly difficult problems devised by Soifer during spring break.

Alexander Soifer

### Backmatter

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