Tham khảo tài liệu 'đề thi toán apmo (châu á thái bình dương)_đề 22', khoa học tự nhiên, toán học phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 2nd United States of America Junior Mathematical Olympiad Day II 12 30 PM - 5 PM EDT April 28 2011 JMO 4. A word is defined as any finite string of letters. A word is a palindrome if it reads the same backwards as forwards. Let a sequence of words W0 W1 W2 . be defined as follows W0 a W1 b and for n 2 Wn is the word formed by writing Wn_2 followed by Wn_ 1. Prove that for any n 1 the word formed by writing W1 W2 . Wn in succession is a palindrome. JMO 5. Points A B C D E lie on circle and point P lies outside the circle. The given points are such that i lines PB and PD are tangent to ii P A C are collinear and iii DE II AC. Prove that BE bisects AC. JMO 6. Consider the assertion that for each positive integer n 2 the remainder upon dividing 22 by 2n 1 is a power of 4. Either prove the assertion or find with proof a counterexample. Copyright Committee on the American Mathematics Competitions Mathematical Association of .
