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The good, the bad and the pleasure (not pressure!) of mathematics competitionsSIU Man Keung ()University of Hong Kong

mathsiu@hkucc.hku.hkThe good of mathematics competitionsclear and logical presentationtenacity and assiduityacademic sincerityThe good of mathematics competitionsclear and logical presentationtenacity and assiduityacademic sincerityThe good of mathematics competitionsclear and logical presentationtenacity and assiduityacademic sincerityThe good of mathematics competitionsclear and logical presentationtenacity and assiduityacademic sincerityarouse a passion for and pique the interest in mathematicsThe bad of mathematics competitionscompetition problems vs researchover-training?Etvs Mathematics Competition in Hungary (started in 1894)

Almost all notable Hungarian mathematicians were winners in the competition when they were school pupils, such as Lipt Fejr, Theodore von Krmn, Denes Knig, Alfrd Haar, Marcel Riesz, George Plya, Gbor Szeg, Tibor Rad, Paul Erds, Paul Turn,

Note the role played by Kzpiskolai Matematikai s Fizikai Lapok, a journal on mathematics and physics for secondary schools founded in 1894. IMO MedalistsBla Bollobs (Hungary, 1959B, 1960G, 1961G) Senior Whitehead Prize, etc.Grigori Margulis (USSR, 1962S) Fields Medalist, Wolf Prize, etc.Lszl Lovsz (Hungary, 1962S, 1965G + Special Prize, 1966G + Special Prize)Fulkerson Prize, Wolf Prize, Kyoto Prize, etc.Richard Ewen Borchards (UK, 1977S, 1978G) Fields Medalist, etc.Peter Shor (USA, 1977S) Navanlinna Prize, MacArthur Award, etc.Noam D. Elkies (USA, 1981G, 1982G) Putnam fellow, etc.William Timothy Gowers (UK, 1981G) Fields Medalist, etc.Grigori Perelman (USSR, 1982G) Fields Medalist (declined), Millennium Prize (declined), etc. Terence Tao (Australia, 1986B, 1987S, 1988G) Fields Medalist, Crafoord Prize, MacArthur Award, etc.

Putnam Fellows in the William Lowell Putnam Mathematical CompetitionRichard Feynman (1939)Nobel Laureate in Physics, etc.

John Milnor (1949,1950)Fields Medalist, Wolf Prize, Abel Prize, etc.

David Mumford (1955,1956)Fields Medalist, Shaw Prize, Wolf Prize, etc.

Daniel Grey Quillen (1959)Fields Medalist, etc.

Peter Shor (1978)Navanlinna Prize, MacArthur Award, etc.

. . . . . . .The bad of mathematics competitionscompetition problems vs researchover-training?Is the passion for the subject of mathematics itself genuine? Is the interest sustained?

Rewrite the equation as a2 k ab + b2 = k ----- (*).If (a, b) is an integral solution of (*), then ab 0, or else ab + 1 0 so that a2 + b2 = (ab + 1) k 0, which is a contradiction. Furthermore a > 0, b > 0, or else k is a perfect square !Let (a, b) be an integral solution of (*) with a > 0, b > 0 and a + b smallest. May assume a > b. We shall produce from it another integral solution (a, b) of (*) with a > 0, b > 0 and a + b < a + b. This is a contradiction! Regard (*) as a quadratic equation with roots a, a , then a + a = kb and aa = b2 k.Show that a is a positive integer and (a, b) is a solution of (*). Note that a = (b2 k) / a (b2 1) / a (a2 1) / a < a, so a + b < a + b .

My false insight: a = N3, b = N. a2 + b2 = N2 (N4 + 1) , ab + 1 = N4 +1,

hence

My ill-fated strategy: Try to deduce from a2 + b2 = (ab + 1) k the equality [a (3b2 3b + 1)]2 + [b 1]2 = {[a (3b2 3b + 1)] [b 1] + 1} { k [2b 1]}.

Failed! Brute-force search:

a18273064112125216240343418512b1238430562771128k14941642536949464ALL IS CLEAR !The solutions of a2 + b2 = (ab + 1) k consist of (N3, N, N2) where N {1, 2, 3, } , which is called the Nth basic solution; (a, b, k) obtained from a basic solution via a sequence of transformations of the form ai + 1 = ai ki bi, bi + 1 = ai, ki + 1 = ki.

Barker sequence (R.H. Barker, 1953)

Each off-phase shift and the original sequence differ by at most one place of coinciding entries and non-coinciding entries in their overlapping part, i.e. its aperiodic autocorrelation function has absolute value 0 or 1 at all off-phase values.

Barker sequence (R.H. Barker, 1953)

Examples and non-examples:1 1 1 0 1 is a Barker sequence of length 5.1 1 1 0 1 0 is not a Barker sequence of length 6.

There are Barker sequences of length 2, 3, 4, 5, 7, 11, 13:s=211s=3110s=41110s=511101s=71110010s=1111100010010s=131111100110101Theorem (R. Turyn, J. Storer, 1961)There is NO Barker sequence of odd length s larger than 13.Conjecture (unsettled for over half a century)There is NO Barker sequence of length s larger than 13.The problem stimulates a lot of research in combinatorial design. Researchers change the problem with various variations and obtain fruitful results.Jonathan Jedwab, What can be used instead of a Barker sequence? Contemporary Mathematics, 461 (2008), pp. 153-178.Olympiad mathematicselementary number theoryalgebracombinatoricssequencesinequalitiesfunctional equationsplane and solid geometry

Olympiad mathematicselementary number theoryalgebracombinatoricssequencesinequalitiesfunctional equationsplane and solid geometry

Why cant this type of so-called Olympiad mathematics be made good use of in the classroom of school mathematics as well? John Baylis and Rod Haggarty, Alice in Numberland: A Students Guide to the Enjoyment of Higher Mathematics (1988)

The professional mathematician will be familiar with the idea that entertainment and serious intent are not incompatible: the problem for us is to ensure that our readers will enjoy the entertainment but not miss the mathematical point, .Von Neumann with the first Institute computer

John Von Neumann (1903-1957) Two cyclist A and B at a distance 20 miles apart approached each other, each going at a speed of 10 m.p.h. A bee flew back and forth between A and B at a speed of 15 m.p.h. When A and B met, how far had the bee travelled?Lw BA vBee b

(t1)(t1)(t2)(t3)(t2 + t3)(t1+ t2)

John von Neumann(1903 1957)Positional warfare Guerilla warfare Each approach has its separate merit. They complement and supplement each other.Positional warfare Guerilla warfare Each approach has its separate merit. They complement and supplement each other. Each approach calls for day-to-day preparation and solid basic knowledge.Positional warfare Guerilla warfare Flexibility and spontaneity are called for in positional warfare. Careful prior preparation and groundwork are needed in guerrilla warfare.( )History of Song Dynasty Biography of Yue Fei() Setting up the battle formation is the routine of art of war. Manuvring the battle formation skillfully rest solely with the mind.Yue Fei ()(1103-1142)

Archimedes(287 B.C. 212B.C.)LIU Hui (3rd century)ArchimedesMeasurement of Circle (3rd century B.C.)(100 B.C. 100 A.D.)Commentaries by LIU Hui(3rd century)

(The finer one divides, the smaller is the left over; divide and again divide until one cannot divide further, then the regular polygon coincides with the circle with nothing being left out.)Commentary of LIU Hui to Problem 31, 32 in Chapter 1 of Jiu Zhang Suan Shu (3rd century)

Archimedes On the Quadrature of the Parabola (3rd century B.C.)

the IMO should be seen as just the tip of a very large, more interesting, iceberg, for it should provide an incentive for each country to establish a pyramid of activities for masses of interested students. [what I learnt from my friend Tony Gardiner, an experienced UK IMO leader] Postscript in M.K. Siu, Some reflections of a coordinator on the IMO , Mathematics Competitions, 8 (1) (1995), 73-77.Other activities: mathematics club mathematics magazine problem session contest in doing projects (at various levels, to various depth) contest in writing book reports and essays, producing cartoons, videos, softwares, toys, games, puzzles,

The good, the bad and the pleasure of mathematics competitions Are to which we should pay our attention.Benefit from the good; avoid the bad; And soak in the pleasure.Then we will find for ourselves satisfaction! to the audience for coming to the talk, to the organizers of TAIMC 2012 for inviting me to give the talk, to Ms. Mimi Lui of the HKU Department of Mathematics for assisting me in the preparation of slides.Thank you