MathDB

This ISL 2005 problem has not been used in any TST I know. A pity, since it is a nice problem, but in its shortlist formulation, it is absolutely incomprehensible. Here is a mathematical restatement of the problem:
Let

k

be a nonnegative integer.
A forest consists of rooted (i. e. oriented) trees. Each vertex of the forest is either a leaf or has two successors. A vertex

v

is called an extended successor of a vertex

u

if there is a chain of vertices

u_{0}=u

u_{1}

u_{2}

, ...,

u_{t-1}

u_{t}=v

with

t>0

such that the vertex

u_{i+1}

is a successor of the vertex

u_{i}

for every integer

i

with

0\leq i\leq t-1

. A vertex is called dynastic if it has two successors and each of these successors has at least

k

extended successors.
Prove that if the forest has

n

vertices, then there are at most

\frac{n}{k+2}

dynastic vertices.

2

Problems(2)