Tham khảo tài liệu 'đề thi toán apmo (châu á thái bình dương)_đề 25', khoa học tự nhiên, toán học phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | THE 1989 ASIAN PACIFIC MATHEMATICAL OLYMPIAD Time allowed 4 hours NO calculators are to be used. Each question is worth seven points. Question 1 Let x-1 x2 . xn be positive real numbers and let S X1 X2 xn. Prove that S2 S3 Sn 1 x1 1 x2 1 xn 1 S 2 3 n - Question 2 Prove that the equation 6 6a2 3b2 c2 5n2 has no solutions in integers except a b c n 0. Question 3 Let A1 A2 A3 be three points in the plane and for convenience let A4 A1 A5 A2. For n 1 2 and 3 suppose that Bn is the midpoint of AnAn 1 and suppose that Cn is the midpoint of AnBn. Suppose that AnCn 1 and BnAn 2 meet at Dn and that AnBn 1 and CnAn 2 meet at En. Calculate the ratio of the area of triangle D1D2D3 to the area of triangle E1E2E3. Question 4 Let S be a set consisting of m pairs a b of positive integers with the property that 1 a b n. Show that there are at least 4m m - Ý 3n triples a b c such that a b a c and b c belong to S. Question 5 Determine all functions f from the reals to the reals for which 1 f x is strictly increasing 2 f x g x 2x for all real X where g x is the composition inverse function to f x . Note f and g are said to be composition inverses if f g x x and g f x x for all real x.
