4

Part of 2012 JBMO TST - Turkey

Problems(2)

Symmetric 3-variable inequality

Source: Turkey JBMO Team Selection Test Problem 4

Find the greatest real number

M

a^2+b^2+c^2+3abc \geq M(ab+bc+ca)

for all non-negative real numbers

a,b,c

a+b+c=4.

inequalitiesfunctioninequalities proposed3-variable inequality

Connected simple graph

Source: Turkey JBMO Team Selection Test Problem 8

G

be a connected simple graph. When we add an edge to

G

(between two unconnected vertices), then using at most

17

edges we can reach any vertex from any other vertex. Find the maximum number of edges to be used to reach any vertex from any other vertex in the original graph, i.e. in the graph before we add an edge.

floor functionceiling functioncombinatorics proposedcombinatorics