2

Part of 1985 Greece National Olympiad

Problems(2)

3 acute angles is max for convex n-gon, equilateral 1985 Greece MO X p2

Source:

a) Prove that a convex

n

-gon cannot have more than

3

interior angles acute. b) Prove that a convex

n

3

interior angles equal to

60^0,

is equilateral.

geometryconvex polygonEquilateral

f(x)=x has real solution(s) if f(f(x))=x has real solution(s), f continuous

Source: 1985 Greece MO Grade XII p2

Conside the continuous

f: \mathbb{R}\to\mathbb{R}

. It is also know that equation

f(f(f(x)))=x

has solution in

\mathbb{R}

. Prove that equation

f(x)=x

has solution in

\mathbb{R}

algebracontinuous function