2007 Indonesia MO

Part of Indonesia MO

Subcontests

(8)

1

k^2+2kn+m^2 is a perfect square

m

n

be two positive integers. If there are infinitely many integers

k

such that k^2\plus{}2kn\plus{}m^2 is a perfect square, prove that m\equal{}n.

1

Points, lines, circle

A,B,C,D

S

AB

is the diameter of

S

CD

is not the diameter. Given also that

C

D

are on different sides of

AB

. The tangents of

S

C

D

P

Q

R

are the intersections of line

AC

BD

AD

BC

, respectively. (a) Prove that

P

Q

R

are collinear. (b) Prove that

QR

is perpendicular to line

AB

1

Simultaneous equations of three real numbers

Find all triples

(x,y,z)

of real numbers which satisfy the simultaneous equations
x \equal{} y^3 \plus{} y \minus{} 8
y \equal{} z^3 \plus{} z \minus{} 8
z \equal{} x^3 \plus{} x \minus{} 8.

1

Rooks in a chessboard

r

s

be two positive integers and

P

a 'chessboard' with

r

s

M

denote the maximum value of rooks placed on

P

such that no two of them attack each other. (a) Determine

M

. (b) How many ways to place

M

P

such that no two of them attack each other? [Note: In chess, a rook moves and attacks in a straight line, horizontally or vertically.]

1

Beautiful arrangements

A 10-digit arrangement

0,1,2,3,4,5,6,7,8,9

is called beautiful if (i) when read left to right,

0,1,2,3,4

form an increasing sequence, and

5,6,7,8,9

form a decreasing sequence, and (ii)

0

is not the leftmost digit. For example,

9807123654

is a beautiful arrangement. Determine the number of beautiful arrangements.

1

Three positive real numbers

a,b,c

be positive real numbers which satisfy 5(a^2\plus{}b^2\plus{}c^2)<6(ab\plus{}bc\plus{}ca). Prove that these three inequalities hold: a\plus{}b>c, b\plus{}c>a, c\plus{}a>b.

1

Positive divisors

For every positive integer

n

b(n)

denote the number of positive divisors of

n

p(n)

denote the sum of all positive divisors of

n

. For example, b(14)\equal{}4 and p(14)\equal{}24. Let

k

be a positive integer greater than

1

. (a) Prove that there are infinitely many positive integers

n

which satisfy b(n)\equal{}k^2\minus{}k\plus{}1. (b) Prove that there are finitely many positive integers

n

which satisfy p(n)\equal{}k^2\minus{}k\plus{}1.

1

Points on a triangle

ABC

be a triangle with \angle ABC\equal{}\angle ACB\equal{}70^{\circ}. Let point

D

BC

AD

is the altitude, point

E

AB

such that \angle ACE\equal{}10^{\circ}, and point

F

is the intersection of

AD

CE

. Prove that CF\equal{}BC.