HMMT 二月 2011 · TEAM2 赛 · 第 8 题

HMMT February 2011 — TEAM2 Round — Problem 8

[ 20 ] Let n be a positive integer, and let a , a , . . . , a be a set of positive integers such that a = 2 and 1 2 n 1 n − 1 a = ϕ ( a ) for all 1 ≤ m ≤ n − 1. Prove that a ≥ 2 . m m +1 n Introduction to the Symmedian [70] The problems in this section require complete proofs . If A , B , and C are three points in the plane that do not all lie on the same line, the symmedian from A in triangle ABC is defined to be the reflection of the median from A in triangle ABC about the bisector of angle A . Like the ϕ function, it turns out that the symmedian satisfies some interesting properties, too. For instance, just like how the medians from A , B , and C all intersect at the centroid of triangle ABC , the symmedians from A , B , and C all intersect at what is called (no surprises here) the symmedian point of triangle ABC . The proof of this fact is not easy, but it is unremarkable. In this section, you will investigate some surprising alternative constructions of the symmedian.

解析

[ 20 ] Let n be a positive integer, and let a , a , . . . , a be a set of positive integers such that a = 2 and 1 2 n 1 n − 1 a = ϕ ( a ) for all 1 ≤ m ≤ n − 1. Prove that a ≥ 2 . m m +1 n Solution: From problem six, it follows that a is even for all m ≤ n − 1. From problem seven, it m n − 2 n − 1 follows that a ≥ 2 a for all m ≤ n − 1. We may conclude that a ≥ a ≥ 2 a = 2 , as m m − 1 n n − 1 1 desired. Introduction to the Symmedian [70] The problems in this section require complete proofs . If A , B , and C are three points in the plane that do not all lie on the same line, the symmedian from A in triangle ABC is defined to be the reflection of the median from A in triangle ABC about the bisector of angle A . Like the ϕ function, it turns out that the symmedian satisfies some interesting properties, too. For instance, just like how the medians from A , B , and C all intersect at the centroid of triangle ABC , the symmedians from A , B , and C all intersect at what is called (no surprises here) the symmedian point of triangle ABC . The proof of this fact is not easy, but it is unremarkable. In this section, you will investigate some surprising alternative constructions of the symmedian.