" Đề thi Olympic sinh viên thế giới năm 1997 ngày 1 " . Đây là một sân chơi lớn để sinh viên thế giới có dịp gặp gỡ, trao đổi, giao lưu và thể hiện khả năng học toán, làm toán của mình. Từ đó đến nay, các kỳ thi Olympic sinh viênthế giới đã liên tục được mở rộng quy mô rất lớn. Kỳ thi này là một sự kiện quan trọng đối với phong trào học toán của sinh viên thế giới. | FOURTH INTERNATIONAL COMPETITION FOR UNIVERSITY STUDENTS IN MATHEMATICS July 30 - August 4 1997 Plovdiv BULGARIA First day August 1 1997 Problems and Solutions Problem 1. Let l n g1 1 be a sequence of positive real numbers such that lim n n nn 0. Find lim - V ln f- 3 n 1 n n k 1 where ln denotes the natural logarithm. Solution. It is well known that -1 Z1 J0 lim 1 n ln - I 1 n n k 1 n ik - --- -1. nJ n i Riemman s sums . Then 1 n - ln -k i Given 0 there exist no such that 0 n for all n no. Then 1 f ln - On 1 f ln - . nn nn k 1 X k i Since 1 n lim 1 k i r 1 ln xdx 1 we obtain the result when goes to 0 and so Problem 2. Suppose 52 an converges. Do the following sums have to converge as n i well a ai 0-2 a a 3 ag a ag as aig ais ag b ai a2 as a4 as a ag ag ag aii ais ais a 10 a12 ai4 ai6 ai7 aig Justify your answers. Solution. 1 n a Yes. Let S 52 an Sn 52 ak. Fix 0 and a number n0 such n i k i that Sn S for n n0. The partial sums of the permuted series have the form L2 -i k S2n-i S2n S2 _k 0 k 2n_i and for 2n_i n0 we and L3 2n-2 S2n-i have L2n-i k S 3 . the permuted series converges. - 1 n i b No. Take an .Then L3 2 2 S2 1 2n 2 OO so L3 2n-2 --- OO. y 2n nn nn 2 -i 1 52 k 2n 2 2k 1 Problem 3. Let A and B be real nxn matrices such that A2 B2 AB. Prove that if BA AB is an invertible matrix then n is divisible by 3. Solution. Set S A wB where w - We have SS A wB A B A2 wBA wAB B2 AB wBA wAB w BA - AB because 1 w. Since det SS det S. det S is a real number and det w BA AB wn det BA AB and det BA AB 0 then wn is a real number. This is possible only when n is divisible by 3. 2 Problem 4. Let a be a real number 1 a 2. a Show that a has a unique representation as an infinite product 11 1 - I 1 - 17 2 where each is a positive integer satisfying 2 2 i i b Show that a is rational if and only if its infinite product has the following property For some m and all k m k i n2k Solution. a We construct inductively the sequence i and the ratios dk Q ------f- nk 1 n so that dk 1
