MathDB

2

Part of 2010 Turkey Team Selection Test

Problems(2)

Turkey TST 2010 Q2

Source:

9/1/2010
Show that ∑cyc(a2+b2)(a2−ab+b2)24≤23(a2+b2+c2)(1a+b+1b+c+1c+a) \sum_{cyc} \sqrt[4]{\frac{(a^2+b^2)(a^2-ab+b^2)}{2}} \leq \frac{2}{3}(a^2+b^2+c^2)\left(\frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a}\right) for all positive real numbers a, b, c.a, \: b, \: c.
inequalitiesinequalities proposed
Turkey TST 2010 Q5

Source:

9/1/2010
For an interior point DD of a triangle ABC,ABC, let ΓD\Gamma_D denote the circle passing through the points A, E, D, FA, \: E, \: D, \: F if these points are concyclic where BD∩AC={E}BD \cap AC=\{E\} and CD∩AB={F}.CD \cap AB=\{F\}. Show that all circles ΓD\Gamma_D pass through a second common point different from AA as DD varies.
geometrycircumcircleanalytic geometryparallelogramgeometry proposed