Colorado Mathematical Olympiad

448 Pages·2011·7.89 MB·English

by  Alexander Soifer

The Colorado Mathematical Olympiad
and Further Explorations

Alexander Soifer
The Colorado Mathematical
Olympiad and Further
Explorations
From the Mountains of Colorado
to the Peaks of Mathematics
With over 185 Illustrations
Forewords by
Philip L. Engel
Paul Erdo˝s
Martin Gardner
Branko Gru¨nbaum
Peter D. Johnson, Jr.
and Cecil Rousseau
123
AlexanderSoifer
CollegeofLetters,ArtsandSciences
UniversityofColoradoatColoradoSprings
1420AustinBluffsParkway
ColoradoSprings,CO80918
USA
[email protected]
ISBN:978-0-387-75471-0 e-ISBN:978-0-387-75472-7
DOI:10.1007/978-0-387-75472-7
SpringerNewYorkDordrechtHeidelbergLondon
LibraryofCongressControlNumber:2011925380
(cid:2)c AlexanderSoifer2011
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Toall thosepeople
throughout the world,
whocreate Olympiads
for new generations of mathematicians.
Forewords to The Colorado Mathematical
Olympiad and Further Explorations
We live in an age of extreme specialization – in mathematics
as well as in all other sciences, in engineering, in medicine.
Hence, to say that probably 90% of mathematicians cannot
understand 90% of mathematics currently published is, most
likely,toooptimistic.Incontrast,evenapessimistwouldhave
to agree that at least 90% of the material in this book is read-
ily accessible to, and understandable by 90% of students in
middle and high schools. However, this does not mean that
the topics are trivial – they are elementary in the sense that
they do not require knowledge of lots of previously studied
material, but are sophisticated in requiring attention, concen-
tration, and thinking that is not fettered by preconceptions.
The organization in groups of five problems for each of the
“Olympiads”, for which the participants were allowed four
hours, hints at the difficulty of finding complete solutions.
Iamconvincedthatmostprofessional mathematicianswould
be hard pressed to solve a set of five problems in two hours,
or evenfour.
Therearemanycollectionsofproblems,for“Olympiads”
ofvarious levels,aswell asproblems in avariety of journals.
Whatsetsthisbookapartfromthe“competition”aresev-
eral aspectsthat deserve to benoted.
vi
Forewords vii
(cid:129)
Theserenityandenthusiasmwithwhichtheproblems,and
their solutions, are presented;
(cid:129)
The absence of prerequisites for understanding the prob-
lemsandtheir solutions;
(cid:129)
The mixture of geometric and combinatorial ideas that are
required in almost all cases.
The detailed exposition of the trials and tribulations en-
dured by the author, as well as the support he received, shed
light on the variety of influences which the administration of
a university exerts on the faculty. As some of the negative ac-
tions are very probably a consequence of mathophobia, the
spirit of this book may cure at least a few present or future
decidersfrom that affliction.
Many mathematiciansare certainly able tocome upwith
an interesting elementary problem. But Soifer may be unique
in his persistence, over the decades, of inventing worthwhile
problems, and providing amusing historical and other com-
ments, all accessible to the intendedpre-college students.
It is my fervent hope that this book will find the wide
readershipitdeserves,andthatitsreaderswillfeelmotivated
to look for enjoyment inmathematics.
Branko Gru¨nbaum
Departmerntof Mathematics
UniversityofWashington, Seattle
Here is another wonderful book from Alexander Soifer. This
one is a more-than-doubling of an earlier book on the first
10 years of the Colorado Mathematical Olympiad, which
was founded and nourished to robust young adulthood by–
AlexanderSoifer.
Like The Mathematical Coloring Book, this book is not so
much mathematical literature as it is literature built around
mathematics, if you will permit the distinction. Yes, there is
plenty of mathematics here, and of the most delicious kind.
In case you were unaware of, or had forgotten (as I had), the
level of skill, nay, art, necessary to pose good olympiad- or
Putnam exam-style problems, or the effect that such a prob-
lem can have on a young mind, and even on the thoughts of
a jaded sophisticate, then what you have been missing can be
foundhereinplenty–atleastayear’ssupplyofgreatintellec-
tual gustation. If you are a mathematics educator looking for
activities for a math. club – your search is over! And with the
Further Explorations sections, anyone so inclined could spend
a lifetime on the mathematics sprouting from thisvolume.
Butsincetherewillbenoshortage ofpraiseforthemath-
ematicalandpedagogicalcontributions ofFromtheMountains
ofColorado...,letmeleavethataspectofthework andsupply
viii
Forewords ix
a few words about the historical account that surrounds and
binds the mathematical trove, and makes a story of it all. The
Historical Notes read like a war diary, or an explorer’s let-
ters home: there is a pleasant, mundane rhythm of reportage
– who and how many showed up from where, who the spon-
sors were, which luminaries visited, who won, what their
prizes were – punctuated by turbulence, events ranging from
A. Soifer’s scolding of a local newspaper for printing only
the names of the top winners, to the difficulties arising from
the weather and the shootings at Columbine High School in
1999 (both matters of life and death in Colorado), to the in-
explicable attempts of university administrators to impede,
restructure, banish, or destroy the Colorado Mathematical
Olympiad, in 1985, 1986,2001 and 2003. Itis fascinating stuff.
The very few who have the entrepeneurial spirit to attempt
the creation ofanythinglike anOlympiad will beforewarned
and inspired.
The rest of us will be pleasurably horrified and amazed,
our sympathies stimulated and our support aroused for the
brave ones who bring new life to the communication of
mathematics.
PeterD. Johnson. Jr.
Departmentof Mathematicsand Statistics
Auburn University