tailieunhanh - ĐỀ THI TOÁN APMO (CHÂU Á THÁI BÌNH DƯƠNG)_ĐỀ 29

Tham khảo tài liệu 'đề thi toán apmo (châu á thái bình dương)_đề 29', khoa học tự nhiên, toán học phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | THE 1993 ASIAN PACIFIC MATHEMATICAL OLYMPIAD Time allowed 4 hours NO calculators are to be used. Each question is worth seven points. Question 1 Let ABCD be a quadrilateral such that all sides have equal length and angle ABC is 60 deg. Let l be a line passing through D and not intersecting the quadrilateral except at D . Let E and F be the points of intersection of l with AB and BC respectively. Let M be the point of intersection of CE and AF. Prove that CA2 CM CE. Question 2 Find the total number of different integer values the function f x x 2x 5x 3x 4x 3 takes for real numbers x with 0 x 100. Question 3 Let f x anxn an-1xn 1 a0 and g x Cn ixn 1 Cnxn C0 be non-zero polynomials with real coefficients such that g x x r f x for some real number r. If a max IanI . a0 and c max I cn 11 . C01 prove that a n 1. Question 4 Determine all positive integers n for which the equation xn 2 x n 2 - x n 0 has an integer as a solution. Question 5 Let P1 P2 . P1993 P0 be distinct points in the x -plane with the following properties i both coordinates of Pi are integers for i 1 2 . 1993 ii there is no point other than Pi and Pi 1 on the line segment joining Pi with Pi 1 whose coordinates are both integers for i 0 1 . . . 1992. Prove that for some i 0 i 1992 there exists a point Q with coordinates qx qy on the line segment joining Pi with Pi 1 such that both 2qx and 2qy are odd .