A strage inequality with phi
Source: Oliforum math contest, problem 2
September 30, 2009
inequalitiesinequalities proposed
Problem Statement
Define the positive real root of x^2 \minus{} x \minus{} 1 and let be positive real numbers such that (a \plus{} 2b)^2 \equal{} 4c^2 \plus{} 1.
Show that \displaystyle 2d^2 \plus{} a^2\left(\phi \minus{} \frac {1}{2}\right) \plus{} b^2\left(\frac {1}{\phi \minus{} 1} \plus{} 2\right) \plus{} 2 \ge 4(c \minus{} d) \plus{} 2\sqrt {d^2 \plus{} 2d} and find all cases of equality.
(A.Naskov)