3
Part of 2008 Turkey MO (2nd round)
Problems(2)
Turkey NMO 2008 Problem 3
Source: Turkey NMO 2008 Problem 3
12/1/2008
Let a.b.c be positive reals such that their sum is 1. Prove that
\frac{a^{2}b^{2}}{c^{3}(a^{2}\minus{}ab\plus{}b^{2})}\plus{}\frac{b^{2}c^{2}}{a^{3}(b^{2}\minus{}bc\plus{}c^{2})}\plus{}\frac{a^{2}c^{2}}{b^{3}(a^{2}\minus{}ac\plus{}c^{2})}\geq \frac{3}{ab\plus{}bc\plus{}ac}
inequalitiesLaTeXinequalities unsolved
Turkey 2008 National Mathematical Olympiad 6th Question
Source: A game on a network
12/6/2008
There is a connected network with computers, in which any of the two cycles don't have any common vertex. A hacker and a administrator are playing a game in this network. On the move hacker selects one computer and hacks it, on the move administrator selects another computer and protects it. Then on every 2k\plus{}1th move hacker hacks one more computer(if he can) which wasn't protected by the administrator and is directly connected (with an edge) to a computer which was hacked by the hacker before and on every 2k\plus{}2th move administrator protects one more computer(if he can) which wasn't hacked by the hacker and is directly connected (with an edge) to a computer which was protected by the administrator before for every . If both of them can't make move, the game ends. Determine the maximum number of computers which the hacker can guarantee to hack at the end of the game.
functioncombinatorics unsolvedcombinatorics