MathDB

1

Part of 2010 Serbia National Math Olympiad

Problems(2)

Inequality with n towns and m two way airlines

Source: Serbia NMO 2010 problem 1

3/11/2011
Some of nn towns are connected by two-way airlines. There are mm airlines in total. For i=1,2,⋯ ,ni = 1, 2, \cdots, n, let did_i be the number of airlines going from town ii. If 1≤di≤20101\le d_i \le 2010 for each i=1,2,⋯ ,2010i = 1, 2,\cdots, 2010, prove that \displaystyle\sum_{i=1}^n d_i^2\le 4022m- 2010n Find all nn for which equality can be attained.
Proposed by Aleksandar Ilic
inequalitiesfloor functioncombinatorics
Prove that <APQ = 2<CAP if AQ/QB=DP/DE

Source: Serbia NMO 2010 problem 4

3/11/2011
Let OO be the circumcenter of triangle ABCABC. A line through OO intersects the sides CACA and CBCB at points DD and EE respectively, and meets the circumcircle of ABOABO again at point P≠OP \neq O inside the triangle. A point QQ on side ABAB is such that AQQB=DPPE\frac{AQ}{QB}=\frac{DP}{PE}. Prove that ∠APQ=2∠CAP\angle APQ = 2\angle CAP.
Proposed by Dusan Djukic
geometrycircumcirclegeometric transformationreflectiontrigonometryincentergeometry unsolved