MathDB

2

Part of 1989 IberoAmerican

Problems(2)

Trigonometric inequality for non-obtuse angles

Source: IberoAmerican 1989 Q2

11/27/2010
Let x,y,zx,y,z be real numbers such that 0≤x,y,z≤π20\le x,y,z\le\frac{\pi}{2}. Prove the inequality π2+2sin⁡xcos⁡y+2sin⁡ycos⁡z≥sin⁡2x+sin⁡2y+sin⁡2z.\frac{\pi}{2}+2\sin x\cos y+2\sin y\cos z\ge\sin 2x+\sin 2y+\sin 2z.
inequalitiestrigonometryinequalities proposed
Function defined on N with conditions

Source: IberoAmerican 1989 Q5

11/27/2010
Let the function ff be defined on the set N\mathbb{N} such that
\text{(i)}\ \   f(1)=1 \text{(ii)}\   f(2n+1)=f(2n)+1 \text{(iii)}  f(2n)=3f(n)
Determine the set of values taken ff.
functionPutnamalgebra proposedalgebra