MathDB

2

Part of 2010 Portugal MO

Problems(2)

Easy line segment lenght - Portuguese MO, Problem 2, 2010

Source:

8/28/2010
On a circumference, points AA and BB are on opposite arcs of diameter CDCD. Line segments CECE and DFDF are perpendicular to ABAB such that A−E−F−BA-E-F-B (i.e., AA, EE, FF and BB are collinear on this order). Knowing AE=1AE=1, find the length of BFBF.
geometry proposedgeometry
Triangle sides quotient - Portuguese MO, Problem 5, 2010

Source:

8/28/2010
Show that any triangle has two sides whose lengths aa and bb satisfy 5−12<ab<5+12\frac{\sqrt{5}-1}{2}<\frac{a}{b}<\frac{\sqrt{5}+1}{2}.
geometry proposedgeometry