3

Part of 1996 Turkey MO (2nd round)

Problems(2)

Integers on real axis

Source: Turkish NMO 1996, 3. Problem

n

integers on the real axis be colored. Determine for which positive integers

k

there exists a family

K

of closed intervals with the following properties: i) The union of the intervals in

K

contains all of the colored points; ii) Any two distinct intervals in

K

are disjoint; iii) For each interval

I

K

{{a}_{I}}=k.{{b}_{I}}

{{a}_{I}}

denotes the number of integers in

I

{{b}_{I}}

the number of colored integers in

I

combinatorics proposedcombinatorics

Proving existence of x,y such that f(x+y)<=f(x)(1+yf(x))

Source: Turkish NMO 1996, 6. Problem

Show that there is no function

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

f(x+y)>f(x)(1+yf(x))

x,y\in {{\mathbb{R}}^{+}}

functioninductionalgebra proposedalgebra