360 problems for mathematical contests

'I'D ANDREESCD DORIN ANDRICA 360 Problems for Mathematical Contests [...]... suggest that they try to find their own solutions to the problems before reading the given ones Many problems can be solved in multiple ways and pertain to interesting extensions This edition is significantly different from the 2002 Romanian edition It features more recent problems, enhanced solutions, along with references for all published problems We wish to extend our gratitude to everyone who... theory, because many contest problems require knowledge in this field The comprehensive problems in the last chapter are also intended to help undergraduate students participating in mathematics contests hone their problem solving skills Students and teachers can find here ideas and questions that can be interesting topics for mathematics circles Due to the difficulty level of the problems contained in this... numerous contest committees All the featured problems are supposed to be original They are the fruit of our collaboration for the last 30 years with several elementary mathematics journals from all over the world Many of these problems were used in contests throughout these years, from the first round to the international level It is possible that some problems are already known, but this is not critical... 2 k=l for all positive integers n 38 Let Xn 2n = 2 + 1, n = 1,2,3, Prove that 1 2 Xl X2 2n - 22 X3 l 1 -+-+-+ + 0 such that = 2f(xy) for all... ai+1), i=l 26 Let m and n be positive integers Prove that xmn -1 xn-l ->-m x for any positive real number x 27 Prove that m! ;::: (n!)[~] for all positive integers m and n 28 Prove that 1 1 1 1+-+-+'''+->n ~2 -v'2 v'3 \Iii n +1 for any integer n ;::: 2 29 Prove that n (1- 1/ vn) + 1 > 1 + ~ + ~ + + ! > n 2 3 n (\In + 1- 1) for any positive integer n 30 Let aI, a2, , an E (0,1 ) and let tn na1a2'"... at least a perfect cube for any intege 10 32 Compute the sum n "'" Sn = ~(-1) 1c(Ic+l) 2 k k=l 33 Compute the sums: a) Sn = ~ (k + l)l(k + 2) (~); b) Tn = ~ (k + l)(k + 2)(k + 3) (~) 13 1.1 PROBLEMS 34 Show that for any positive integer n the number Cn; 1)22n + Cn: 1) Cn 22n - 2 3+ + : 1)3n 2 is the sum of two consecutive perfect squares 35 Evaluate the sums: 36 Prove that 12 for all integers n ~ n(n... important thing is that an educated - to a certain extent - reader will find in this book problems that bring something new and will teach new ways of dealing with key mathematics concepts, a variety of methods, tactics, and strategies The problems are divided in chapters, although this division is not firm, for some of the problems require background in several fields of mathematics Besides the traditional... properties f(l) f: a1 and f(k) f: a1 For f(l) and f(k) there are n - 1 possibilities of choosing a character from C2, •.• ,Cn and for f(i), 1 < i < k there are n such possibilities Therefore the number of strings f(l)f(2) f(k - l)f(k) is Nk = (n - 1)2n k- 2 It follows that N1 + N2 + + N m = (n - 1) + (n - 1)2nO + (n - 1)2n 1 + + (n - 1)2n m - 2 = (Dorin Andrica) 2 Suppose, for the sake of contradiction,... 2(1978), pp 75, Problem 3698) 3 Note that (1) 17 1 ALGEBRA 18 and furthermore (2) for all integers ml, m2 2:: 1 Suppose that for all integers k 2:: 1 we have I[k](x) f= x Because there are n! permutations, it follows that for k positive integers nl > n2 such that > n! there are distinct (3) Let h = nl - n2 > 0 and observe that for all k the functions I[k] are injective, since numbers ai, i = 1, n are distinct... intended to help students preparing for all rounds of Mathematical Olympiads or any other significant mathematics contest Teachers will also find this work useful in training young talented students Our experience as contestants was a great asset in preparing this book To this we added our vast personal experience from the other side of the" barricade" , as creators of problems and members of numerous . ANDREESCU DORIN ANDRICA 360 Problems for Mathematical Contests GIL Publishing House © GIL Publishing House ISBN 973-9417-12-4 360 Problems for Mathematical Contests Authors: Titu Andreescu,. 360 Problems for Mathematical Contests