Combinatorial Problems in Mathematical Competitions by Yao Zhang PDF

By Yao Zhang


This e-book makes a speciality of combinatorial difficulties in mathematical competitions. It offers easy wisdom on how one can resolve combinatorial difficulties in mathematical competitions, and in addition introduces very important ideas to combinatorial difficulties and a few regular issues of often-used ideas. a few enlightening and novel examples and routines are good selected during this e-book.
With this ebook, readers can discover, research and summarize the tips and techniques of fixing combinatorial difficulties. Their mathematical tradition and talent might be greater remarkably after interpreting this ebook.

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Additional info for Combinatorial Problems in Mathematical Competitions (Mathematical Olympiad)

Sample text

D;, is valid, where {d ,,} satisfies the recurrence relation ® . With the mathematical induction, we prove that this conjecture is true. For n = °and n = 1, all = 1 = d~, al = 4 = df. Assume that for n = k - 1 and n = k , we have that aH = d~-I , ak = d~. Then for n = k + 1, we get a k+1 = 14ak - ak-I - 6 14d~ - d ~ -1 - = 6 = (4d k -d k_I )2 - 2(d~ + d~-I - 4d k d k - 1 + 3) = d ~ +1 -2(d~ + d~-1 -4d kd k - 1 + 3). But d~ + d~-I - 4dkdk-l +3 = d k C4d k - 1 - dk-2) + d~ - 1 - 4d k d k- 1 +3 = d~-1 -d k-2 C4d k- 1 -d k- 2 ) +3 = d~- I + d~ - 2 -4dk-ldk-2 + 3 = df = 22 + d6 -4d 1d u +3 + 12 - 4 X 1X2 +3 = 0.

A" = d;, is a perfect square. Remark The recurrence relation of undetermined coefficients. ® is also deduced by the method In fact, setting d ,,+2 = pd ,,+1 + qd" and combining this with d o d 1 = 2, d 2 = 7, d 3 = 26, we get q {P=4, { 2P + =7, 7 P + 2q = 26. ::::;. q = - 1. Therefored"+2 =4d,,+1 - d"Cn = 0, 1, 2, ... ).

Consider the triangle BCD, if a side of D BCD, say BD, is colored red, then the 3 sides of D ABD are all colored red. Otherwise, the 3 sides of D BCD all are colored blue. This completes the proof. It seems convenient to list some basic concepts of the graph theory that will be used throughout the book. The set V" of n points in a 22 Combinatorial Problems in Mathematical Competitions plane is called the set of vertices. A complete graph on n vertices, denoted by K" , consists of a set V" of vertices and a set E of edges (or sides) connecting each pair of distinct vertices in V.

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