" Đề thi Olympic sinh viên thế giới năm 2005 " . 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 đại. | 12th International Mathematics Competition for University Students Blagoevgrad July 22 - July 28 2005 First Day Problem 1. Let A be the n x n matrix whose i j th entry is i j for all i j 1 2 . n. What is the rank of A Solution 1. For n 1 the rank is 1. Now assume n 2. Since A i nj i j nj i matrix A is the sum of two matrixes of rank 1. Therefore the rank of A is at most 2. The determinant of the top-left 2 x 2 minor is -1 so the rank is exactly 2. Therefore the rank of A is 1 for n 1 and 2 for n 2. Solution 2. Consider the case n 2. For i n n 1 . 2 subtract the i 1 th row from the nth row. Then subtract the second row from all lower rows. 2 3 . . n U 2 3 . . n 1 1 1 0 . 2 . 1 . 0 . .n .1 .0 . rank 3 . . 4 . . . n 2 . . . rank 1 . . 1 . . . .1 . . . rank 2 . n 1 n 2 . . . 2n . 1 1 . . .1 . . 0 . 0 . . . .0 Problem 2. For an integer n 3 consider the sets Sn xi X2 . Xn Vi Xi G 0 1 2 An X1 X2 . . . Xn G Sn Vi n 2 x xi i xi 2 1 and Bn xi X2 . Xn G Sn Vi n 1 Xi Xi 1 Xi 0 . Prove that An i 3 Bn . A denotes the number of elements of the set A. Solution 1. Extend the definitions also for n 1 2. Consider the following sets An xi X2 . . . Xn G An Xn-1 Xn A An An Bn Xi X2 . Xn G Bn Xn 0 Bn Bn Bn and denote an Ara a n An a n A bn Bra bn B bn B . It is easy to observe the following relations between the a-sequences an an 1 o an 1 an an an 2an 2an which lead to an 1 2an 2an-1. For the b-sequences we have the same relations bn bn 1 b bn 1 bn bn b n 2bn 2bn therefore bn i 2bn 2bn-i. By computing the first values of an and bn we obtain o1 3 o2 9 a3 24 b1 3 b2 8 12th International Mathematics Competition for University Students Blagoevgrad July 22 - July 28 2005 Second Day Problem 1. Let f x x2 bx c where b and c are real numbers and let M x G R f x 1 . Clearly the set M is either empty or consists of disjoint open intervals. Denote the sum of their lengths by M . Prove that M 2 2. Solution. Write f x x 2 d where d c If d 1 then f x 1 for all x M 0 and M If 1 d 1 then f x 1 for all x
