MathDB

2

Part of 2008 239 Open Mathematical Olympiad

Problems(2)

Equal radii lengths

Source: 239 2008 S2

7/28/2020
A circumscribed quadrilateral ABCDABCD is given. EE and FF are the intersection points of opposite sides of the ABCDABCD. It turned out that the radii of the inscribed circles of the triangles AEFAEF and CEFCEF are equal. Prove that AC⊥BDAC \bot BD.
geometry
1/a + 1/b + 1/c = 1 Inequality

Source: 239 2008 J2

7/28/2020
For all positive numbers a,b,ca, b, c satisfying 1a+1b+1c=1\frac{1}{a} + \frac{1}{b} + \frac{1}{c} = 1, prove that: aa+bc+bb+ca+cc+ab≥34. \frac{a}{a+bc} + \frac{b}{b+ca} + \frac{c}{c+ab} \geq \frac{3}{4} .
inequalitiesalgebra