2008 Indonesia MO

Part of Indonesia MO

Subcontests

(4)

1

Power and divisibility

m,n > 1

are integers which satisfy n|4^m \minus{} 1 and 2^m|n \minus{} 1. Is it a must that n \equal{} 2^{m} \plus{} 1?

1

Natural number expression

Find all natural number which can be expressed in \frac{a\plus{}b}{c}\plus{}\frac{b\plus{}c}{a}\plus{}\frac{c\plus{}a}{b} where

a,b,c\in \mathbb{N}

satisfy \gcd(a,b)\equal{}\gcd(b,c)\equal{}\gcd(c,a)\equal{}1

2

It got a number of solutions..

Prove that for x,y\in\mathbb{R^ \plus{} }, \frac {1}{(1 \plus{} \sqrt {x})^{2}} \plus{} \frac {1}{(1 \plus{} \sqrt {y})^{2}} \ge \frac {2}{x \plus{} y \plus{} 2}

One implementation of graph theory

In a group of 21 persons, every two person communicate with different radio frequency. It's possible for two person to not communicate (means there's no frequency occupied to connect them). Only one frequency used by each couple, and it's unique for every couple. In every 3 persons, exactly two of them is not communicating to each other. Determine the maximum number of frequency required for this group. Explain your answer.

2

Identity of a subset

Let A \equal{} \{1,2,\ldots,2008\} a) Find the number of subset of

A

which satisfy : the product of its elements is divisible by 7 b) Let

N(i)

denotes the number of subset of

A

which sum of its elements remains

i

when divided by 7. Prove that N(0) \minus{} N(1) \plus{} N(2) \minus{} N(3) \plus{} N(4) \minus{} N(5) \plus{} N(6)\minus{}N(7) \equal{} 0 EDITED : thx for cosinator.. BTW, your statement and my correction give 80% hint of the solution :D

N to N function

Find all function

f: \mathbb{N}\rightarrow\mathbb{N}

satisfy f(mn)\plus{}f(m\plus{}n)\equal{}f(m)f(n)\plus{}1 for all natural number

n