tailieunhanh - ĐỀ THI TOÁN APMO (CHÂU Á THÁI BÌNH DƯƠNG)_ĐỀ 28

Tham khảo tài liệu 'đề thi toán apmo (châu á thái bình dương)_đề 28', khoa học tự nhiên, toán học phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | THE 1992 ASIAN PACIFIC MATHEMATICAL OLYMPIAD Time allowed 4 hours NO calculators are to be used. Each question is worth seven points. Question 1 A triangle with sides a b and c is given. Denote by s the semiperimeter that is s a b c 2. Construct a triangle with sides s a s b and s c. This process is repeated until a triangle can no longer be constructed with the side lengths given. For which original triangles can this process be repeated indefinitely Question 2 In a circle C with centre O and radius r let C1 C2 be two circles with centres Cl O2 and radii r1 r2 respectively so that each circle Ci is internally tangent to C at Ai and so that C1 C2 are externally tangent to each other at A. Prove that the three lines OA O1A2 and O2A1 are concurrent. Question 3 Let n be an integer such that n 3. Suppose that we choose three numbers from the set 1 2 . n . Using each of these three numbers only once and using addition multiplication and parenthesis let us form all possible combinations. a Show that if we choose all three numbers greater than n 2 then the values of these combinations are all distinct. b Let p be a prime number such that p ựn. Show that the number of ways of choosing three numbers so that the smallest one is p and the values of the combinations are not all distinct is precisely the number of positive divisors of p 1. Question 4 Determine all pairs h s of positive integers with the following property If one draws h horizontal lines and another s lines which satisfy i they are not horizontal ii no two of them are parallel iii no three of the h s lines are concurrent then the number of regions formed by these h s lines is 1992. Question 5 Find a sequence of maximal length consisting of non-zero integers in which the sum of any seven consecutive terms is positive and that of any eleven consecutive terms is .