tailieunhanh - ĐỀ THI TOÁN APMO (CHÂU Á THÁI BÌNH DƯƠNG)_ĐỀ 30

Tham khảo tài liệu 'đề thi toán apmo (châu á thái bình dương)_đề 30', khoa học tự nhiên, toán học phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | THE 1994 ASIAN PACIFIC MATHEMATICAL OLYMPIAD Time allowed 4 hours NO calculators are to be used. Each question is worth seven points. Question 1 Let f R R be a function such that i For all x y 2 R f x f y 1 f x y f x f y ii For aU X 2 0 Q f 0 f x iii -f -1 f 1 1 Find all such functions f. Question 2 Given a nondegenerate triangle ABC with circumcentre O orthocentre H and circumradius R prove that OHI 3R. Question 3 Let n be an integer of the form a2 b2 where a and b are relatively prime integers and such that if p is a prime p ựn then p divides ab. Determine all such n. Question 4 Is there an infinite set of points in the plane such that no three points are collinear and the distance between any two points is rational Question 5 You are given three lists A B and C. List A contains the numbers of the form 10fc in base 10 with k any integer greater than or equal to 1. Lists B and C contain the same numbers translated into base 2 and 5 respectively A 10 100 1000 BC 1010 20 1100100 400 1111101000 13000 Prove that for every integer n 1 there is exactly one number in exactly one of the lists B or C that has exactly n .