MathDB

3

Part of 2001 Romania Team Selection Test

Problems(3)

Inequality for sides of a triangle a,b,c

Source: Romanian TST 2001

1/16/2011
The sides of a triangle have lengths a,b,ca,b,c. Prove that: \begin{align*}(-a+b+c)(a-b+c)\, +\, & (a-b+c)(a+b-c)+(a+b-c)(-a+b+c)\\ &\le\sqrt{abc}(\sqrt{a}+\sqrt{b}+\sqrt{c})\end{align*}
inequalitiesinequalities proposed
Two rays among n form acute angle

Source: Romanian TST 2001

1/16/2011
Find the least n∈Nn\in N such that among any nn rays in space sharing a common origin there exist two which form an acute angle.
vectorinductionlinear algebracombinatorics proposedcombinatorics
Tangents to the circumcircle of ABC

Source:

1/16/2011
The tangents at AA and BB to the circumcircle of the acute triangle ABCABC intersect the tangent at CC at the points DD and EE, respectively. The line AEAE intersects BCBC at PP and the line BDBD intersects ACAC at RR. Let QQ and SS be the midpoints of the segments APAP and BRBR respectively. Prove that ∠ABQ=∠BAS\angle ABQ=\angle BAS.
geometrycircumcirclegeometry proposed