1

Part of 2016 International Zhautykov Olympiad

Problems(2)

Source: Izho 2016 day 1 p1

A quadrilateral

ABCD

is inscribed in a circle with center

O

. It's diagonals meet at

M

.The circumcircle of

ABM

intersects the sides

AD

BC

N

K

respectively. Prove that areas of

NOMD

KOMC

geometrycircumcircle

Functional inequality

Source: IZHO2016

k>0

for which a strictly decreasing function

g:(0;+\infty)\to(0;+\infty)

exists such that

g(x)\geq kg(x+g(x))

for all positive

x