MathDB

Macedonia National Olympiad 2017 Problem 5

Source: Macedonia National Olympiad 2017

April 8, 2017

functionalgebra

Problem Statement

Let

n>1 \in \mathbb{N}

and

a_1, a_2, ..., a_n

be a sequence of

n

natural integers. Let:

b_1 = \left[\frac{a_2 + \cdots + a_n}{n-1}\right], b_i = \left[\frac{a_1 + \cdots + a_{i-1} + a_{i+1} + \cdots + a_n}{n-1}\right], b_n = \left[\frac{a_1 + \cdots + a_{n-1}}{n-1}\right]

Define a mapping

f

by

f(a_1,a_2, \cdots a_n) = (b_1,b_2,\cdots,b_n)

.
a) Let

g: \mathbb{N} \to \mathbb{N}

be a function such that

g(1)

is the number of different elements in

f(a_1,a_2, \cdots a_n)

and

g(m)

is the number od different elements in

f^m(a_1,a_2, \cdots a_n) = f(f^{m-1}(a_1,a_2, \cdots a_n)); m>1

. Prove that

\exists k_0 \in \mathbb{N}

s.t. for

m \ge k_0

the function

g(m)

is periodic.
b) Prove that

\sum_{m=1}^k \frac{g(m)}{m(m+1)} < C

for all

k \in \mathbb{N}

, where

C

is a function that doesn't depend on

k

.

Back to Problems View on AoPS