MathDB

2

Part of 2009 Turkey MO (2nd round)

Problems(2)

Turkey NMO 2009 Q2

Source:

8/31/2010
Let Γ\Gamma be the circumcircle of a triangle ABC,ABC, and let DD and EE be two points different from the vertices on the sides ABAB and AC,AC, respectively. Let A′A' be the second point where Γ\Gamma intersects the bisector of the angle BAC,BAC, and let PP and QQ be the second points where Γ\Gamma intersects the lines A′DA'D and A′E,A'E, respectively. Let RR and SS be the second points of intersection of the lines AA′AA' and the circumcircles of the triangles APDAPD and AQE,AQE, respectively. Show that the lines DS, ERDS, \: ER and the tangent line to Γ\Gamma through AA are concurrent.
geometrycircumcirclegeometric transformationtrapezoidhomothetygeometry proposed
Turkey NMO 2009 Q5

Source:

8/31/2010
Show that (b+c)(a4−b2c2)ab+2bc+ca+(c+a)(b4−c2a2)bc+2ca+ab+(a+b)(c4−a2b2)ca+2ab+bc≥0 \frac{(b+c)(a^4-b^2c^2)}{ab+2bc+ca}+\frac{(c+a)(b^4-c^2a^2)}{bc+2ca+ab}+\frac{(a+b)(c^4-a^2b^2)}{ca+2ab+bc} \geq 0 for all positive real numbers a, b, c.a, \: b , \: c.
inequalitiescalculusderivativefunctioninequalities proposed