tailieunhanh - ĐỀ THI TOÁN APMO (CHÂU Á THÁI BÌNH DƯƠNG)_ĐỀ 32

Tham khảo tài liệu 'đề thi toán apmo (châu á thái bình dương)_đề 32', khoa học tự nhiên, toán học phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | THE 1996 ASIAN PACIFIC MATHEMATICAL OLYMPIAD Time allowed 4 hours NO calculators are to be used. Each question is worth seven points. Question 1 Let ABCD be a quadrilateral AB BC CD DA. Let MN and PQ be two segments perpendicular to the diagonal BD and such that the distance between them is d BD 2 with M 2 AD N 2 DC P 2 AB and Q 2 BC. Show that the perimeter of hexagon AMNCQP does not depend on the position of MN and PQ so long as the distance between them remains constant. Question 2 Let m and n be positive integers such that n m. Prove that 2nn m n m2 m n In _ m 11 m m n Question 3 Let Pl P2 P3 P4 be four points on a circle and let I1 be the incentre of the triangle P2P3P4 I2 be the incentre of the triangle P1P3P4 I3 be the incentre of the triangle P1P2P4 I4 be the incentre of the triangle P1P2P3. Prove that I1 I2 I3 I4 are the vertices of a rectangle. Question 4 The National Marriage Council wishes to invite n couples to form 17 discussion groups under the following conditions 1. All members of a group must be of the same sex . they are either all male or all female. 2. The difference in the size of any two groups is 0 or 1. 3. All groups have at least 1 member. 4. Each person must belong to one and only one group. Find all values of n n 1996 for which this is possible. Justify your answer. Question 5 Let a b c be the lengths of the sides of a triangle. Prove that pa b c pb c a pc a b pa pb pc and determine when equality .