TAILIEUCHUNG - Đề thi toán khu vực 1999-2009

Tài liệu tham khảo, bao gồm các bài tập dạng tự luận, được viết bằng tiếng anh, các bạn có thể học sâu hơn khi giải các bài này, tài liệu hay và bổ các bạn học tốt | MathLInks Vietnam Team Selection Tests 1999 Day 1 T Let an odd prime p be a given number satisfying 2h 1 mod p for all h p 1 h E N p p . Let us consider the sequence an n 0 and an even integer a E defined by a0 a and an 1 p bn for n 0 1 2 . where bn is the greatest odd divisor of an. Show that an is periodical and find its least positive period. 2 Two polynomials f x and g x with real coefficients are called similar if there exist nonzero real number a such that f x q g x for all x E R. I. Show that there exists a polynomial P x of degree 1999 with real coefficients which satisfies the condition P x 2 4 and Pf x 2 x2 4 are similar. II. How many polynomials of degree 1999 are there which have above mentioned property. 3 Let a convex polygon H be given. Show that for every real number a E 0 1 there exist 6 distinct points on the sides of H denoted by A1 A2 . A6 clockwise satisfying the conditions I. AiA2 A5A4 a A6A3 . II. Lines A1A2 A5A4 are equidistant from A6A3. By AB we denote vector AB This file was downloaded from the AoPS MathLinks Math Olympiad Resources Page http Page 1 http MathLInks Vietnam Team Selection Tests 1999 Day 2 0 Let a sequence of positive reals un be given. For every positive integer n let kn be the least positive integer satisfying kn n EI ä E -i 1 i 1 Show that the sequence n f has finite limit if and only if un does. kn 2 Let a triangle ABC inscribed in circle r be given. Circle lies in angle of triangle and touches sides AB AC at M1 N1 and touches internally r at P1. The points M2 N2 P2 and M3 N3 P3 are defined similarly to angles B and C respectively. Show that MiN1 M2N2 and M3N3 intersect each other at their midpoints. 3 Let a regular polygon with p vertices be given where p is an odd prime number. At every vertex there is one monkey. An owner of monkeys takes p peanuts goes along the perimeter of polygon clockwise and delivers to the monkeys by the following rule Gives the first peanut for the

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