3

Part of 2008 Bulgaria Team Selection Test

Problems(2)

Find all real numbers

Source: Bulgarian IMO TST 2008, Day 1, Problem 3

\mathbb{R}^{+}

be the set of positive real numbers. Find all real numbers

a

for which there exists a function

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

3(f(x))^{2}=2f(f(x))+ax^{4}

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

functionalgebra proposedalgebra

Digraph with infinitely many vertices

Source: Bulgarian IMO TST 2008, Day 2, Problem 3

G

be a directed graph with infinitely many vertices. It is known that for each vertex the outdegree is greater than the indegree. Let

O

be a fixed vertex of

G

. For an arbitrary positive number

n

V_{n}

be the number of vertices which can be reached from

O

passing through at most

n

O

counts). Find the smallest possible value of

V_{n}

floor functionceiling functioninductionalgebrapolynomialcombinatorics proposedcombinatorics