MathDB

2

Part of 2012 South East Mathematical Olympiad

Problems(2)

inscribed circle

Source: China south east mathematical Olympiad 2012 day1 problem 2

7/17/2013
The incircle II of △ABC\triangle ABC is tangent to sides AB,BC,CAAB,BC,CA at D,E,FD,E,F respectively. Line EFEF intersects lines AI,BI,DIAI,BI,DI at M,N,KM,N,K respectively. Prove that DM⋅KE=DN⋅KFDM\cdot KE=DN\cdot KF.
geometrycircumcircletrigonometrygeometry unsolved
find the least natural number

Source: China south east mathematical olympiad 2012 day2 problem 6

7/17/2013
Find the least natural number nn, such that the following inequality holds:n−20112012−n−20122011<n−201320113−n−201120133\sqrt{\dfrac{n-2011}{2012}}-\sqrt{\dfrac{n-2012}{2011}}<\sqrt[3]{\dfrac{n-2013}{2011}}-\sqrt[3]{\dfrac{n-2011}{2013}}.
inequalitiesinequalities unsolved