3

Part of 2005 Iran MO (2nd round)

Problems(2)

M has at least 79 members - Iran NMO 2005 - Problem3

Source:

In one galaxy, there exist more than one million stars. Let

M

be the set of the distances between any

2

of them. Prove that, in every moment,

M

79

members. (Suppose each star as a point.)

geometry3D geometryspherecombinatorics proposedcombinatorics

Find all functions (x+y)f(f(x)y)=x^2f(f(x)+f(y)) - Iran NMO 2005 - Problem6

Source: Iran NMO 2005 - Problem 6

Find all functions

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

such that for all positive real numbers

x

y

, the following equation holds:

(x+y)f(f(x)y)=x^2f(f(x)+f(y)).

functionalgebra proposedalgebrafunctional equation