MathDB

2

Part of 1998 China National Olympiad

Problems(2)

Do there exist such 2n positive integers?

Source: Chinese Mathematical Olympiad 1998 Problem 2

8/22/2013
Given a positive integer n>1n>1, determine with proof if there exist 2n2n pairwise different positive integers a1,…,an,b1,…bna_1,\ldots ,a_n,b_1,\ldots b_n such that a1+…+an=b1+…+bna_1+\ldots +a_n=b_1+\ldots +b_n and n−1>∑i=1nai−biai+bi>n−1−11998.n-1>\sum_{i=1}^{n}\frac{a_i-b_i}{a_i+b_i}>n-1-\frac{1}{1998}.
inequalities unsolvedinequalities
Determine the position of D from condition

Source: China mathematical olympiad 1998 problem5

8/22/2013
Let DD be a point inside acute triangle ABCABC satisfying the condition DA⋅DB⋅AB+DB⋅DC⋅BC+DC⋅DA⋅CA=AB⋅BC⋅CA.DA\cdot DB\cdot AB+DB\cdot DC\cdot BC+DC\cdot DA\cdot CA=AB\cdot BC\cdot CA. Determine (with proof) the geometric position of point DD.
inequalitiesgeometryparallelogramgeometry unsolved