2

Part of 2007 Bulgaria Team Selection Test

Problems(2)

Functional Inequality

Source: Bulgarian TST 2007 for Balkan MO and ARO, I day Problem 2

a\in\mathbb{R}

for which there exists a non-constant function

f: (0,1]\rightarrow\mathbb{R}

a+f(x+y-xy)+f(x)f(y)\leq f(x)+f(y)

x,y\in(0,1].

inequalitiesfunctionalgebra proposedalgebra

k-element subsets = (2n-k)*(2-element)

Source: Bulgarian TST 2007 for Balkan MO and ARO, II day Problem 2

n,k

be positive integers such that

n\geq2k>3

A= \{1,2,...,n\}.

n

k

such that the number of

k

-element subsets of

A

2n-k

times bigger than the number of

2

-element subsets of

A.

inequalitiesnumber theory proposednumber theory