tailieunhanh - ĐỀ THI TOÁN APMO (CHÂU Á THÁI BÌNH DƯƠNG)_ĐỀ 5

Tham khảo tài liệu 'đề thi toán apmo (châu á thái bình dương)_đề 5', khoa học tự nhiên, toán học phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | APMO XVII Asian Pacific Mathematics Olympiad Time allowed 4 hours Each problem is worth 7 points The contest problems are to be kept confidential until they are posted on the official APMO website. Please do not disclose nor discuss the problems over the internet until that date. No calculators are to be used during the contest. Problem 1. Prove that for every irrational real number a there are irrational real numbers b and b0 so that a b and ab0 are both rational while ab and a b0 are both irrational. Problem 2. Let a b and c be positive real numbers such that abc 8. Prove that a2 b2 c2 4 P 1 a3 1 b3 P 1 b3 1 c3 P 1 c3 1 a3 - 3 Problem 3. Prove that there exists a triangle which can be cut into 2005 congruent triangles. Problem 4. In a small town there are n n houses indexed by i j for 1 i j n with 1 1 being the house at the top left corner where i and j are the row and column indices respectively. At time 0 a hre breaks out at the house indexed by 1 c where c n. During each subsequent time interval t t 1 the hre hghters defend a house which is not yet on hre while the hre spreads to all undefended neighbors of each house which was on hre at time t. Once a house is defended it remains so all the time. The process ends when the hre can no longer spread. At most how many houses can be saved by the hre hghters A house indexed by i j is a neighbor of a house indexed by k Ế if i k j 1. Problem 5. In a triangle ABC points M and N are on sides AB and AC respectively such that MB BC CN. Let R and r denote the circumradius and the inradius of the triangle ABC respectively. Express the ratio MN BC in terms of R and .