24th Chinese Mathematical Olympiad (Continued)

The 24th Chinese Mathematical Olympiad (CMO) was held in Hainan province in January 2009. There are total 184 high school students from all over China participated in this contest. There are 11 students got all points (126 points), 46 students received gold medal (for score higher than or equal to 93 points, 75 students received silver medal (for score higher than or equal to 51 points, 77 students received bronze medal (for score higher than or equal to 6 points).

In a previous post, I translated the questions in the test during the first day. Here are the question during the second day on January 10, 2009 from 8:00 a.m. to 12:30 p.m.

4. Given an integer n ≥ 3, and real numbers a₁, a₂,…, a_n such that min_1≤i<j≤n|a_i – a_j| = 1. Find the minimum value of ∑_k=1ⁿ|a_k|³.

5. Each side and each diagonal of a convex n-polygon is painted in a color out of n different colors. Question: For what kind of n, there is a way to paint it such that for any three colors in the given n colors, there can be found a triangle for which all its vertices are the vertices of the polygon and the three sides are painted in these three colors.

6. Given an integer n ≥ 3, show that there is a set of n distinct positive integers and for any two non-empty subsets A, B, the numbers

∑_x∈Ax/|A| and ∑_x∈Bx/|B|

are coprime composite numbers. (Here ∑_x∈Xx and |X| represent the sum of values of all elements in a finite set X and the number of elements respectively.)

Note: This is my personal translation and I do not guarantee they are 100% correct. I do not have solutions to these problems.

Tags: China, education, event, geometry, integer, math, mathematics, number, olympiad, polygon