2

Part of 2009 International Zhautykov Olympiad

Problems(2)

Beautiful functional equation-inequality, ZIMO-09

Source: International Zhautykov Olympiad 2009, day 1, problem 2

a

, such that there exist a function

f: \mathbb{R}\rightarrow\mathbb{R}

satisfying the following inequality: x\plus{}af(y)\leq y\plus{}f(f(x)) for all

x,y\in\mathbb{R}

inequalitiesfunctionalgebra proposedalgebra

Quadrilateral with two opposite angles equal 90, ZIMO - 09

Source: International Zhautykov Olympiad 2009, day 2, problem 5.

Given a quadrilateral

ABCD

with \angle B\equal{}\angle D\equal{}90^{\circ}. Point

M

is chosen on segment

AB

so taht AD\equal{}AM. Rays

DM

CB

intersect at point

N

H

K

are feet of perpendiculars from points

D

C

AC

AN

, respectively. Prove that \angle MHN\equal{}\angle MCK.

geometrycyclic quadrilateralsimilar trianglesgeometry proposed