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Tham khảo tài liệu 'đề thi toán apmo (châu á thái bình dương)_đề 9', khoa học tự nhiên, toán học phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | XIX Asian Pacific Mathematics Olympiad Time allowed 4 hours Each problem is worth 7 points The contest problems are to be kept confidential until they are posted on the official APMO website. Please do not disclose nor discuss the problems over the internet until that date. No calculators are to be used during the contest. Problem 1. Let S be a set of 9 distinct integers all of whose prime factors are at most 3. Prove that S contains 3 distinct integers such that their product is a perfect cube. Problem 2. Let ABC be an acute angled triangle with ABAC 60 and AB AC. Let I be the incenter and H the orthocenter of the triangle ABC. Prove that 2 AHI 3 ABC. Problem 3. Consider n disks Cl C2 . Cn in a plane such that for each 1 i n the center of Ci is on the circumference of Ci 1 and the center of Cn is on the circumference of C1. Define the score of such an arrangement of n disks to be the number of pairs i j for which Ci properly contains Cj. Determine the maximum possible score. Problem 4. Let x y and z be positive real numbers such that ựx py ffiz 1. Prove that x2 yz y2 zx z2 xy 1 p2x2 y z p2y2 z x p2z2 x y Problem 5. A regular 5 X 5 -array of lights is defective so that toggling the switch for one light causes each adjacent light in the same row and in the same column as well as the light itself to change state from on to off or from off to on. Initially all the lights are switched off. After a certain number of toggles exactly one light is switched on. Find all the possible positions of this light.
