MathDB

2

Part of 2003 Moldova Team Selection Test

Problems(3)

Three variables cyclic inequality

Source: Moldova IMO-BMO TST 2003, day 1, problem 2.

8/14/2008
The positive reals x,y x,y and z z are satisfying the relation x \plus{} y \plus{} z\geq 1. Prove that: \frac {x\sqrt {x}}{y \plus{} z} \plus{} \frac {y\sqrt {y}}{z \plus{} x} \plus{} \frac {z\sqrt {z}}{x \plus{} y}\geq \frac {\sqrt {3}}{2} Proposer: Baltag Valeriu
inequalitiesLaTeXinequalities unsolved
Geometric inequality involving lenghts of bisectors

Source: Moldova IMO-BMO TST 2003, day 2, problem 2.

8/15/2008
Consider the triangle ABC ABC with side-lenghts equal to a,b,c a,b,c. Let p\equal{}\frac{a\plus{}b\plus{}c}{2}, R R-the radius of circumcircle of the triangle ABC ABC, r r-the radius of the incircle of the triangle ABC ABC and let la,lb,lc l_a,l_b,l_c be the lenghts of bisectors drawn from A,B A,B and C C, respectively, in the triangle ABC ABC. Prove that: l_al_b\plus{}l_bl_c\plus{}l_cl_a\leq p\sqrt{3r^2\plus{}12Rr} Proposer: Baltag Valeriu
inequalitiesgeometrycircumcircleinequalities proposed
Minimum and maximum value of a sum

Source: Moldova IMO-BMO TST 2003, day 3, problem 2.

8/16/2008
Let a1,a2,...,a2003≥0 a_1,a_2,...,a_{2003}\geq 0, such that a_1\plus{}a_2\plus{}...\plus{}a_{2003}\equal{}2 and a_1a_2\plus{}a_2a_3\plus{}...\plus{}a_{2003}a_1\equal{}1. Determine the minimum and maximum value of a_1^2\plus{}a_2^2\plus{}...\plus{}a_{2003}^2.