3

Part of 2008 Iran MO (2nd Round)

Problems(2)

Iran NMO 2008 (Second Round) - Problem3

Source:

a,b,c,

d

be real numbers such that at least one of

c

d

is non-zero. Let

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

be a function defined as

f(x)=\frac{ax+b}{cx+d}

. Suppose that for all

x\in\mathbb{R}

f(x) \neq x

. Prove that if there exists some real number

a

f^{1387}(a)=a

x

in the domain of

f^{1387}

f^{1387}(x)=x

. Notice that in this problem,

f^{1387}(x)=\underbrace{f(f(\cdots(f(x)))\cdots)}_{\text{1387 times}}.

Hint. Prove that for every function

g(x)=\frac{sx+t}{ux+v}

, if the equation

g(x)=x

2

g(x)=x

x\in\mathbb{R}-\left\{\frac{-v}{u}\right\}

functionalgebra proposedalgebra

Iran NMO 2008 (Second Round) - Problem6

Source:

ABC

H

is the foot of perpendicular from

A

BC

O

is the circumcenter of

\Delta ABC

T,T'

are the feet of perpendiculars from

H

AB,AC

, respectively. We know that

AC=2OT

AB=2OT'

geometrycircumcircletrigonometry