MathDB

2

Part of 2006 IberoAmerican

Problems(2)

Problem 2

Source: 21-st Iberoamerican Mathematical Olympiad

5/6/2007
For n real numbers a1, a2, … ,an,a_{1},\, a_{2},\, \ldots\, , a_{n}, let dd denote the difference between the greatest and smallest of them and S=∑i<j∣ai−aj∣.S = \sum_{i<j}\left |a_i-a_j \right|. Prove that (n−1)d≤S≤n24d(n-1)d\le S\le\frac{n^{2}}{4}d and find when each equality holds.
inequalitiesinequalities unsolved
Problem 5

Source: 21-st Iberoamerican Mathematical Olympiad

5/6/2007
The sides ADAD and CDCD of a tangent quadrilateral ABCDABCD touch the incircle φ\varphi at PP and Q,Q, respectively. If MM is the midpoint of the chord XYXY determined by φ\varphi on the diagonal BD,BD, prove that ∠AMP=∠CMQ.\angle AMP = \angle CMQ.
geometrycyclic quadrilateralprojective geometrygeometry unsolved