HMMT 二月 2011 · TEAM2 赛 · 第 8 题

英文解析

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
   Solution: From problem six, it follows that a is even for all m ≤ n − 1. From problem seven, itmn − 2 n − 1
   follows that a ≥ 2 a for all m ≤ n − 1. We may conclude that a ≥ a ≥ 2 a = 2 , asm 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 thebisector of angle A . Like the ϕ function, it turns out that the symmedian satisfies some interestingproperties, too. For instance, just like how the medians from A , B , and C all intersect at the centroidof 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. Inthis section, you will investigate some surprising alternative constructions of the symmedian.