3

Part of 2016 Azerbaijan Team Selection Test

Problems(2)

Functional Equation

Source: Azerbaijan IMO TST 2016,D2 P3

Prove that there does not exist a function

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

f(f(x)+y)=f(x)+3x+yf(y)

for all positive reals

x,y

functionfunctional equationalgebra

Source: Azerbaijan IMO TST 2016, D3 P3

2016

customers visited the store. Every customer has been only once at the store(a customer enters the store,spends some time, and leaves the store). Find the greatest integer

k

that makes the following statement always true.
We can find

k

customers such that either all of them have been at the store at the same time, or any two of them have not been at the same store at the same time.