2012 Indonesia MO

Part of Indonesia MO

Subcontests

(4)

1

No such function exists

\mathbb{R}^+

be the set of all positive real numbers. Show that there is no function

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

f(x+y)=f(x)+f(y)+\dfrac{1}{2012}

for all positive real numbers

x

y

.
Proposer: Fajar Yuliawan

2

Maximum Number of Shadow

2012

distinct points

A_1,A_2,\dots,A_{2012}

on the Cartesian plane. For any permutation

B_1,B_2,\dots,B_{2012}

A_1,A_2,\dots,A_{2012}

define the shadow of a point

P

as follows: Point

P

180^{\circ}

B_1

P_1

P_1

180^{\circ}

B_2

P_2

P_{2011}

180^{\circ}

B_{2012}

P_{2012}

P_{2012}

is called the shadow of

P

with respect to the permutation

B_1,B_2,\dots,B_{2012}

N

be the number of different shadows of

P

up to all permutations of

A_1,A_2,\dots,A_{2012}

. Determine the maximum value of

N

.
Proposer: Hendrata Dharmawan

B,P,Q,R are collinear

Given a triangle

ABC

, let the bisector of

\angle BAC

BC

and circumcircle of triangle

ABC

D

E

, respectively. Let

M

N

be the midpoints of

BD

CE

, respectively. Circumcircle of triangle

ABD

AN

Q

. Circle passing through

A

that is tangent to

BC

D

AM

AC

respectively at

P

R

. Show that the four points

B,P,Q,R

lie on the same line.
Proposer: Fajar Yuliawan

2

AR perpendicular to BC

Given an acute triangle

ABC

AB>AC

that has circumcenter

O

BO

CO

meet the bisector of

\angle BAC

P

Q

, respectively. Moreover, line

BQ

CP

R

AR

is perpendicular to

BC

.
Proposer: Soewono and Fajar Yuliawan

condition of sqrt(x)+sqrt(y)=sqrt(n) having solution

n

be a positive integer. Show that the equation

\sqrt{x}+\sqrt{y}=\sqrt{n}

have solution of pairs of positive integers

(x,y)

n

is divisible by some perfect square greater than

1

.
Proposer: Nanang Susyanto

2

Even non-negative integer

Show that for any positive integers

a

b

n=\mathrm{LCM}(a,b)+\mathrm{GCD}(a,b)-a-b

is an even non-negative integer.
Proposer: Nanang Susyanto

Same matrices

Given positive integers

m

n

P

Q

be two collections of

m \times n

0

1

m

n

columns. An example of such collections for

m=3

n=4

\left[ \begin{array}{cccc} 1 & 1 & 1 & 0 \\ 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array} \right].

Let those two collections satisfy the following properties: (i) On each row of

P

, from left to right, the numbers are non-increasing, (ii) On each column of

Q

, from top to bottom, the numbers are non-increasing, (iii) The sum of numbers on the row in

P

equals to the same row in

Q

, (iv) The sum of numbers on the column in

P

equals to the same column in

Q

. Show that the number on row

i

j

P

equals to the number on row

i

j

Q

i=1,2,\dots,m

j=1,2,\dots,n

.
Proposer: Stefanus Lie