Tham khảo tài liệu 'đề thi toán apmo (châu á thái bình dương)_đề 31', khoa học tự nhiên, toán học phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | THE 1995 ASIAN PACIFIC MATHEMATICAL OLYMPIAD Time allowed 4 hours NO calculators are to be used. Each question is worth seven points. Question 1 Determine all sequences of real numbers a-1 a2 . a1995 which satisfy 2pan n 1 an 1 n 1 for n 1 2 . 1994 and 2ặ ai995 1994 ai 1. Question 2 Let a1 a2 . an be a sequence of integers with values between 2 and 1995 such that i Any two of the aỉs are realtively prime ii Each ai is either a prime or a product of primes. Determine the smallest possible values of n to make sure that the sequence will contain a prime number. Question 3 Let PQRS be a cyclic quadrilateral such that the segments PQ and RS are not parallel. Consider the set of circles through P and Q and the set of circles through R and S. Determine the set A of points of tangency of circles in these two sets. Question 4 Let C be a circle with radius R and centre O and S a fixed point in the interior of C. Let AA and BB0 be perpendicular chords through S. Consider the rectangles SAMB SBN0A Q A M R and w IV 4 r ind llic cot r t all nninfc A t ỈV M and N 4 arrrnnd an . in te set o a points an wen moves aroun the whole circle. Question 5 Find the minimum positive integer k such that there exists a function f from the set Z of all integers to 1 2 . .kg with the property that f x f y whenever x yj 2 5 7 .
