MathDB

3

Part of 2002 Turkey Team Selection Test

Problems(2)

Cutting a seq. into two s.t. |Sum_1 - Sum_2| <= |some term|

Source: Turkey TST 2002 - P3

4/6/2013
A positive integer nn and real numbers a1,…,ana_1,\dots, a_n are given. Show that there exists integers mm and kk such that ∣∑i=1mai−∑i=m+1nai∣≤∣ak∣.|\sum\limits_{i=1}^m a_i -\sum\limits_{i=m+1}^n a_i | \leq |a_k|.
inequalities proposedinequalities
Connected Monochromatic k-subset

Source: Turkey TST 2002 - P6

4/6/2013
Consider 2n+12n+1 points in space, no four of which are coplanar where n>1n>1. Each line segment connecting any two of these points is either colored red, white or blue. A subset MM of these points is called a connected monochromatic subset, if for each a,b∈Ma,b \in M, there are points a=x0,x1,…,xl=ba=x_0,x_1, \dots, x_l = b that belong to MM such that the line segments x0x1,x1x2,…,xl−1x1x_0x_1, x_1x_2, \dots, x_{l-1}x_1 are all have the same color. No matter how the points are colored, if there always exists a connected monochromatic k−k-subset, find the largest value of kk. (l>1l > 1)
combinatorics proposedcombinatorics