Tham khảo tài liệu 'đề thi toán apmo (châu á thái bình dương)_đề 2', khoa học tự nhiên, toán học phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | XIV Asian Pacific Mathematics Olympiad March 2002 Time allowed 4 hours No calculators are to be used Each question is worth 7 points Problem 1. Let ai a2 a3 . an be a sequence of non-negative integers where n is a positive integer. Let ai a2 an An . n Prove that aiW .an _AnJ n where LAn_l is the greatest integer less than or equal to An and a 1 X 2 X X a for a 1 and 0 1 . When does equality hold Problem 2. Find all positive integers a and b such that a2 b b2 a and b2 a a2 b are both integers. Problem 3. Let ABC be an equilateral triangle. Let P be a point on the side AC and Q be a point on the side AB so that both triangles ABP and ACQ are acute. Let R be the orthocentre of triangle ABP and S be the orthocentre of triangle ACQ. Let T be the point common to the segments BP and CQ. Find all possible values of zCBP and zBCQ such that triangle TRS is equilateral. Problem 4. Let x y z be positive numbers such that 1 1 1 1. X y z Show that px yz ựy zx ựz xy pxyz px py pz. Problem 5. Let R denote the set of all real numbers. Find all functions f from R to R satisfying i there are only finitely many s in R such that f s 0 and ii f x4 y x3f x f f y for all x y in .
