MathDB

3

Part of 2007 Turkey MO (2nd round)

Problems(2)

Turkey NMO 2007 Problem 3, inequality with a+b+c=3

Source: Turkey NMO 2007 Problem 3

9/27/2011
If a,b,ca,b,c are three positive real numbers such that a+b+c=3a+b+c=3, prove that a2+3b2ab2(4−ab)+b2+3c2bc2(4−ab)+c2+3a2ca2(4−ca)≥4 {\frac{a^{2}+3b^{2}}{ab^{2}(4-ab)}}+{\frac{b^{2}+3c^{2}}{bc^{2}(4-ab)}}+{\frac{c^{2}+3a^{2}}{ca^{2}(4-ca)}}\geq 4
inequalitiesinequalities proposed
Turkey NMO 2007 Problem 6, k-directionally connected

Source: Turkey NMO 2007 Problem 6

10/2/2011
In a country between each pair of cities there is at most one direct road. There is a connection (using one or more roads) between any two cities even after the elimination of any given city and all roads incident to this city. We say that the city AA can be k -directionally connected to the city BB, if : we can orient at most kk roads such that after arbitrary orientation of remaining roads for any fixed road ll (directly connecting two cities) there is a path passing through roads in the direction of their orientation starting at AA, passing through ll and ending at BB and visiting each city at most once. Suppose that in a country with nn cities, any two cities can be k - directionally connected. What is the minimal value of kk?
combinatorics proposedcombinatorics