MathDB

5

Part of 2013 Saint Petersburg Mathematical Olympiad

Problems(2)

St.Peterburg, P5 Grade 11, 2013

Source:

4/17/2014
Let x1x_1, ... , xn+1∈[0,1]x_{n+1} \in [0,1] and x1=xn+1x_1=x_{n+1} . Prove that ∏i=1n(1−xixi+1+xi2)≥1. \prod_{i=1}^{n} (1-x_ix_{i+1}+x_i^2)\ge 1.
A. Khrabrov, F. Petrov
inequalitiesinductioninequalities proposed
St.Peterburg, P5 Grade 10, 2013

Source:

4/27/2014
Given quadrilateral ABCDABCD with AB=BC=CDAB=BC=CD. Let AC∩BD=OAC\cap BD=O, X,YX,Y are symmetry points of OO respect to midpoints of BCBC, ADAD, and ZZ is intersection point of lines, which perpendicular bisects of ACAC, BDBD. Prove that X,Y,ZX,Y,Z are collinear.
symmetrygeometry proposedgeometry