2005 Indonesia MO

Part of Indonesia MO

Subcontests

(8)

1

Get acquainted in a competition

90

contestants in a mathematics competition. Each contestant gets acquainted with at least

60

other contestants. One of the contestants, Amin, state that at least four contestants have the same number of new friends. Prove or disprove his statement.

1

Convex Quadrilateral

ABCD

be a convex quadrilateral. Square

AB_1A_2B

is constructed such that the two vertices

A_2,B_1

is located outside

ABCD

. Similarly, we construct squares

BC_1B_2C

CD_1C_2D

DA_1D_2A

K

be the intersection of

AA_2

BB_1

L

be the intersection of

BB_2

CC_1

M

be the intersection of

CC_2

DD_1

N

be the intersection of

DD_2

AA_1

KM

is perpendicular to

LN

1

Three Integers

Find all triples

(x,y,z)

of integers which satisfy x(y \plus{} z) \equal{} y^2 \plus{} z^2 \minus{} 2 y(z \plus{} x) \equal{} z^2 \plus{} x^2 \minus{} 2 z(x \plus{} y) \equal{} x^2 \plus{} y^2 \minus{} 2.

1

Floor function

For an arbitrary real number

x

\lfloor x\rfloor

denotes the greatest integer not exceeding

x

. Prove that there is exactly one integer

m

which satisfy \displaystyle m\minus{}\left\lfloor \frac{m}{2005}\right\rfloor\equal{}2005.

1

Point M in Triangle ABC

M

be a point in triangle

ABC

such that \angle AMC\equal{}90^{\circ}, \angle AMB\equal{}150^{\circ}, \angle BMC\equal{}120^{\circ}. The centers of circumcircles of triangles

AMC,AMB,BMC

P,Q,R

, respectively. Prove that the area of

\triangle PQR

is greater than the area of

\triangle ABC

1

Rational, then Integer

k

m

be positive integers such that \displaystyle\frac12\left(\sqrt{k\plus{}4\sqrt{m}}\minus{}\sqrt{k}\right) is an integer. (a) Prove that

\sqrt{k}

is rational. (b) Prove that

\sqrt{k}

is a positive integer.

1

Equation of integers

For an arbitrary positive integer

n

p(n)

as the product of the digits of

n

(in decimal). Find all positive integers

n

such that 11p(n)\equal{}n^2\minus{}2005.

1

Triangle with Integral Side Lengths

n

be a positive integer. Determine the number of triangles (non congruent) with integral side lengths and the longest side length is

n