MathDB

2

Part of 2010 Serbia National Math Olympiad

Problems(2)

Product of n numbers in scattered cells gives same residue.

Source: Serbia NMO 2010 problem 5

3/11/2011
An n\times n table whose cells are numerated with numbers 1,2,⋯ ,n21, 2,\cdots, n^2 in some order is called Naissus if all products of nn numbers written in nn scattered cells give the same residue when divided by n2+1n^2+1. Does there exist a Naissus table for (a)n=8;(a) n = 8; (b)n=10?(b) n = 10? (nn cells are scattered if no two are in the same row or column.)
Proposed by Marko Djikic
linear algebramatrixcombinatorics unsolvedcombinatorics
Prove that <HMA=<GNS

Source: Serbia NMO 2010 problem 2

3/11/2011
In an acute-angled triangle ABCABC, MM is the midpoint of side BCBC, and D,ED, E and FF the feet of the altitudes from A,BA, B and CC, respectively. Let HH be the orthocenter of ΔABC\Delta ABC, SS the midpoint of AHAH, and GG the intersection of FEFE and AHAH. If NN is the intersection of the median AMAM and the circumcircle of ΔBCH\Delta BCH, prove that ∠HMA=∠GNS\angle HMA = \angle GNS.
Proposed by Marko Djikic
geometrycircumcircleratiogeometric transformationreflectiongeometry unsolved