MathDB

1

Indonesia National Science Olympiad 2010 - Day 2 Problem 6

n>1

such that

\tau(n)+\phi(n)=n+1

Which in this case,

\tau(n)

represents the amount of positive divisors of

n

, and

\phi(n)

represents the amount of positive integers which are less than

n

and relatively prime with

n

.
Raja Oktovin, Pekanbaru

4

1

Indonesia National Science Olympiad 2010 - Day 1 Problem 4

Given that

m

and

n

are positive integers with property:

(mn)\mid(m^{2010}+n^{2010}+n)

Show that there exists a positive integer

k

such that

n=k^{2010}

Nanang Susyanto, Yogyakarta

5

1

Indonesia National Science Olympiad 2010 - Day 2 Problem 5

m

boys and

n

girls (

m>n

) sat across a round table, supervised by a teacher, and they did a game, which went like this. At first, the teacher pointed a boy to start the game. The chosen boy put a coin on the table. Then, consecutively in a clockwise order, everyone did his turn. If the next person is a boy, he will put a coin to the existing pile of coins. If the next person is a girl, she will take a coin from the existing pile of coins. If there is no coin on the table, the game ends. Notice that depending on the chosen boy, the game could end early, or it could go for a full turn. If the teacher wants the game to go for at least a full turn, how many possible boys could be chosen?
Hendrata Dharmawan, Boston, USA

3

1

Indonesia National Science Olympiad 2010 - Day 1 Problem 3

A mathematical competition was attended by 120 participants from several contingents. At the closing ceremony, each participant gave 1 souvenir each to every other participants from the same contingent, and 1 souvenir to any person from every other contingents. It is known that there are 3840 souvenirs whom were exchanged. Find the maximum possible contingents such that the above condition still holds?
Raymond Christopher Sitorus, Singapore

7

1

Indonesia National Science Olympiad 2010 - Day 2 Problem 7

Given 2 positive reals

a

and

b

. There exists 2 polynomials

F(x)=x^2+ax+b

and

G(x)=x^2+bx+a

such that all roots of polynomials

F(G(x))

and

G(F(x))

are real. Show that

a

and

b

are more than

6

.
Raja Oktovin, Pekanbaru

1

Indonesia National Science Olympiad 2010 - Day 1 Problem 1

Let

a,b,c

be three different positive integers. Show that the sequence

a+b+c,ab+bc+ca,3abc

could be neither an arithmetic nor geometric progression.
Fajar Yuliawan, Bandung

2010 Indonesia MO

Subcontests

Indonesia National Science Olympiad 2010 - Day 2 Problem 6

Indonesia National Science Olympiad 2010 - Day 1 Problem 4

Indonesia National Science Olympiad 2010 - Day 2 Problem 5

Indonesia National Science Olympiad 2010 - Day 1 Problem 3

Indonesia National Science Olympiad 2010 - Day 2 Problem 7

Indonesia National Science Olympiad 2010 - Day 1 Problem 1