102 Combinatorial Problems: From the Training of the USA IMO - download pdf or read online

By Titu Andreescu

ISBN-10: 0817643176

ISBN-13: 9780817643171

ISBN-10: 0817682228

ISBN-13: 9780817682224

"102 Combinatorial difficulties" contains conscientiously chosen difficulties which have been utilized in the educational and checking out of america overseas Mathematical Olympiad (IMO) workforce. Key good points: * offers in-depth enrichment within the very important parts of combinatorics via reorganizing and adorning problem-solving strategies and techniques * themes contain: combinatorial arguments and identities, producing features, graph thought, recursive relatives, sums and items, likelihood, quantity idea, polynomials, conception of equations, advanced numbers in geometry, algorithmic proofs, combinatorial and complex geometry, useful equations and classical inequalities The ebook is systematically prepared, progressively development combinatorial abilities and methods and broadening the student's view of arithmetic. apart from its useful use in education lecturers and scholars engaged in mathematical competitions, it's a resource of enrichment that's certain to stimulate curiosity in numerous mathematical components which are tangential to combinatorics.

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Keywords » research - Chebyshev platforms - Combinatorial conception - Dynamical structures - Jacobi identities - Multiexponential research - Singular worth decomposition theorems

Additional resources for 102 Combinatorial Problems: From the Training of the USA IMO Team

Example text

An-t = n - 1. Combining the two inequalities above, we have 1) 2(n- 1) >n(n--- - 2 ' orn::::: 4. Hence n = 1, 2, 3, 4 are the only n satisfying the conditions of the problem. Second Solution: (David Vincent) For each n, define the polynomial fn(x) = x(x + x 2) · · · (x + x 2 + ... + xn). It is clear that fn(x) is a 1 + 2 + can write fn(X) = fn,tX ··· + n = n(nit) degree polynomial. We 2 + fn,2X + · · · + fn • n(n+J)X 2 n(ntl) • Then the coefficient of the term xm in fn(x), [xm]Cfn(x)) = fn,m• is equal to the number of ways that m can be written in the form at + · · · + an with at E {1}, a2 E {1, 2}, ...

AIME 1996] A bored student walks down a hall that contains a row of closed lockers, numbered 1 to 1024. He opens the locker numbered 1, and then alternates between skipping and opening each closed locker thereafter. When he reaches the end of the hall, the student turns around and starts back. He opens the first closed locker he encounters, and then alternates between skipping and opening each closed locker thereafter. The student continues wandering back and forth in this manner until every locker is open.

The sum of S' is 61. Hence the set S we seek is a 5-element set with a sum of at least 61. Let S = {a, b, c, d, e} with a < b < c < d < e, and lets denote the sum of S. Then it is clear that d + e ::::: 29 and c ::::: 13. Since there are G) 2-element subsets of S,a+b::::: d+e-10+ 1::::: 20. Hences::::: 20+ 13+29 = 62. Ifc::::: 12, then S ::::: 61; if c = 13, then d = 14 and e = 15. Then s ::::: a+ b + 42. Since 12+ 15 = 13 + 14, b::::: 11. If b::::: 10, thena+b::::: 19 ands::::: 61; if b = 11, then a::::: 8 as 10 + 15 = 11 + 14 and 9 + 15 = 11 + 13, implying that s ::::: 8 + 11 + 42 = 61.

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102 Combinatorial Problems: From the Training of the USA IMO Team by Titu Andreescu


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