MathDB

3

Part of 2005 Poland - Second Round

Problems(2)

Number of segments and triangles inequality

Source:

12/6/2010
In space are given n≥2n\ge 2 points, no four of which are coplanar. Some of these points are connected by segments. Let KK be the number of segments (K>1)(K>1) and TT be the number of formed triangles. Prove that 9T2<2K39T^2<2K^3.
inequalitiescombinatorics proposedcombinatoricsgraph theoryExtremal combinatorics
Poland 2005

Source:

11/17/2010
Prove that if the real numbers a,b,ca,b,c lie in the interval [0,1][0,1], then abc+1+bac+1+cab+1≤2\frac{a}{bc+1}+\frac{b}{ac+1}+\frac{c}{ab+1}\le 2
convex-concave inequalitiesalgebraPolandinequalitiesinequalities proposed