" Đề thi Olympic sinh viên thế giới năm 2003 " . 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 trong trường. | 10th International Mathematical Competition for University Students Cluj-Napoca July 2003 Day 1 1. a Let ai a2 be a sequence of real numbers such that ai 1 and an i an for all n. Prove that the sequence an IT-1 has a finite limit or tends to infinity. 10 points b Prove that for all a 1 there exists a sequence a1 a2 with the same properties such that an lim -1 a. 10 points Solution. a Let bn -----. Then an 1 3an is equivalent to bn 1 bn thus the sequence 3 n-1 1 2 bn is strictly increasing. Each increasing sequence has a finite limit or tends to infinity. b For all a 1 there exists a sequence 1 b1 b2 which converges to a. Choosing an 3 n bn we obtain the required sequence an . 2. Let a1 a2 a51 be non-zero elements of a field. We simultaneously replace each element with the sum of the 50 remaining ones. In this way we get a sequence b1 b51. If this new sequence is a permutation of the original one what can be the characteristic of the field The characteristic of a field is p if p is the smallest positive integer such that x x x 0for any element x -------------------------------------------------------------------V------- p of the field. If there exists no such p the characteristic is 0. 20 points Solution. Let S a1 a2 a51. Then b1 b2 b51 50S. Since b1 b2 b51 is a permutation of a1 a2 a51 we get 50S S so 49S 0. Assume that the characteristic of the field is not equal to 7. Then 49S 0 implies that S 0. Therefore b aj for i 1 2 51 On the other hand bj av j where p e Ss1- Therefore if the characteristic is not 2 the sequence a1 a2 a51 can be partitioned into pairs aj av j of additive inverses. But this is impossible since 51 is an odd number. It follows that the characteristic of the field is 7 or 2. The characteristic can be either 2 or 7. For the case of 7 x1 x51 1 is a possible choice. For the case of 2 any elements can be chosen such that S 0 since then bj aj aj . 3. Let A be an n x n real matrix such that 3A3 A2 A I I is the identity matrix . Show that the sequence
