MathDB

4

Part of 1986 China Team Selection Test

Problems(2)

sum is over the consecutive square of the segments

Source: China TST 1986, problem 4

5/16/2005
Given a triangle ABCABC for which C=90C=90 degrees, prove that given nn points inside it, we can name them P1,P2,…,PnP_1, P_2 , \ldots , P_n in some way such that: ∑k=1n−1(PKPk+1)2≤AB2\sum^{n-1}_{k=1} \left( P_K P_{k+1} \right)^2 \leq AB^2 (the sum is over the consecutive square of the segments from 11 up to n−1n-1). Edited by orl.
inductiongeometrygeometry unsolved
China TST 1986 4k circle markers

Source: China TST 1986, problem 8

5/16/2005
Mark 4⋅k4 \cdot k points in a circle and number them arbitrarily with numbers from 11 to 4⋅k4 \cdot k. The chords cannot share common endpoints, also, the endpoints of these chords should be among the 4⋅k4 \cdot k points.
i. Prove that 2⋅k2 \cdot k pairwisely non-intersecting chords can be drawn for each of whom its endpoints differ in at most 3⋅k−13 \cdot k - 1. ii. Prove that the 3⋅k−13 \cdot k - 1 cannot be improved.
combinatorics unsolvedcombinatoricspointsChordsChinaTSTTeam Selection Test