2013 Indonesia MO

Part of Indonesia MO

Subcontests

(8)

1

Balanced sets

A

be a set of positive integers.

A

is called "balanced" if [and only if] the number of 3-element subsets of

A

whose elements add up to a multiple of

3

is equal to the number of 3-element subsets of

A

whose elements add up to not a multiple of

3

.
a. Find a 9-element balanced set. b. Prove that no set of

2013

elements can be balanced.

1

Keep expanding squares outside

ABCD

be a parallelogram. Construct squares

ABC_1D_1, BCD_2A_2, CDA_3B_3, DAB_4C_4

on the outer side of the parallelogram. Construct a square having

B_4D_1

as one of its sides and it is on the outer side of

AB_4D_1

and call its center

O_A

. Similarly do it for

C_1A_2, D_2B_3, A_3C_4

O_B, O_C, O_D

AO_A = BO_B = CO_C = DO_D

1

Strong numbers

A positive integer

n

is called "strong" if there exists a positive integer

x

x^{nx} + 1

is divisible by

2^n

.
a. Prove that

2013

is strong. b. If

m

is strong, determine the smallest

y

m

y^{my} + 1

is divisible by

2^m

1

Sum of three square polynomials

P

be a quadratic (polynomial of degree two) with a positive leading coefficient and negative discriminant. Prove that there exists three quadratics

P_1, P_2, P_3

P(x) = P_1(x) + P_2(x) + P_3(x)

P_1, P_2, P_3

have positive leading coefficients and zero discriminants (and hence each has a double root) - The roots of

P_1, P_2, P_3

1

Sum of products of three integers

p > 3

is a prime number and

S = \sum_{2 \le i < j < k \le p-1} ijk

S+1

is divisible by

p

1

Find M so one number exceeds the target

Determine all positive real

M

such that for any positive reals

a,b,c

, at least one of

a + \dfrac{M}{ab}, b + \dfrac{M}{bc}, c + \dfrac{M}{ca}

is greater than or equal to

1+M

1

Circumcircle intersections everywhere

ABC

be an acute triangle and

\omega

be its circumcircle. The bisector of

\angle BAC

\omega

at [another point]

M

P

AM

\triangle ABC

. Lines passing

P

that are parallel to

AB

AC

BC

E, F

respectively. Lines

ME, MF

\omega

K, L

respectively. Prove that

AM, BL, CK

are concurrent.

1

Counting parallelograms

4 \times 6

grid, all edges and diagonals are drawn (see attachment). Determine the number of parallelograms in the grid that uses only the line segments drawn and none of its four angles are right.