tailieunhanh - ĐỀ THI TOÁN APMO (CHÂU Á THÁI BÌNH DƯƠNG)_ĐỀ 33

Tham khảo tài liệu 'đề thi toán apmo (châu á thái bình dương)_đề 33', khoa học tự nhiên, toán học phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | THE 1997 ASIAN PACIFIC MATHEMATICAL OLYMPIAD Time allowed 4 hours NO calculators are to be used. Each question is worth seven points. Question 1 Given S 1 1 1 1 Ĩ Ĩ 1 1 1 1 1 1 . 3 3 6 3 6 1 1993006 where the denominators contain partial sums of the sequence of reciprocals of triangular numbers . k n n 1 2 for n 1 2 . 1996 . Prove that S 1001. Question 2 Find an integer n where 100 n 1997 such that 2n 2 n is also an integer. Question 3 Let ABC be a triangle inscribed in a circle and let la Mpi M M where ma mb mc are the lengths of the angle bisectors internal to the triangle and Ma Mb Mc are the lengths of the angle bisectors extended until they meet the circle. Prove that la 1 lc 3. sin2 A sin2 B sin2 C 3. and that equality holds iff ABC is an equilateral triangle. Question 4 Triangle A1A2A3 has a right angle at A3. A sequence of points is now defined by the following iterative process where n is a positive integer. From An n 3 a perpendicular line is drawn to meet An_2An_1 at An 1. a Prove that if this process is continued indefinitely then one and only one point P is interior to every triangle An_2An_1 An n 3. b Let A1 and A3 be fixed points. By considering all possible locations of A2 on the plane find the locus of P. Question 5 Suppose that n people A1 A2 . An n 3 are seated in a circle and that Ai has ai objects such that ai a- --- an nN where N is a positive integer. In order that each person has the same number of objects each person Aị is to give or to receive a certain number of objects to or from its two neighbours Ai-1 and Ai 1. Here An 1 means Al and An means Ao. How should this redistribution be performed so that the total number of objects transferred is minimum