3

Part of 2010 Turkey MO (2nd round)

Problems(2)

Turkey NMO 2010 P3

Source:

Prove that for all

n \in \mathbb{Z^+}

and for all positive real numbers satisfying

a_1a_2...a_n=1

\displaystyle\sum_{i=1}^{n} \frac{a_i}{\sqrt{{a_i}^4+3}} \leq \frac{1}{2}\displaystyle\sum_{i=1}^{n} \frac{1}{a_i}

inequalities proposedinequalities

Turkey NMO 2010 P6

Source:

K

be the set of all sides and diagonals of a convex

2010-gon

in the plane. For a subset

A

K,

if every pair of line segments belonging to

A

intersect, then we call

A

as an intersecting set. Find the maximum possible number of elements of union of two intersecting sets.

inequalitiesanalytic geometryemailcombinatorics proposedcombinatorics