MathDB

2

Problem 3, Iberoamerican Olympiad 2011

ABC

be a triangle and

X,Y,Z

be the tangency points of its inscribed circle with the sides

BC, CA, AB

, respectively. Suppose that

C_1, C_2, C_3

are circle with chords

YZ, ZX, XY

, respectively, such that

C_1

and

C_2

intersect on the line

CZ

and that

C_1

and

C_3

intersect on the line

BY

. Suppose that

C_1

intersects the chords

XY

and

ZX

at

J

and

M

, respectively; that

C_2

intersects the chords

YZ

and

XY

at

L

and

I

, respectively; and that

C_3

intersects the chords

YZ

and

ZX

at

K

and

N

, respectively. Show that

I, J, K, L, M, N

lie on the same circle.

Problem 6, Iberoamerican Olympiad 2011

Let

k

and

n

be positive integers, with

k \geq 2

. In a straight line there are

kn

stones of

k

colours, such that there are

n

stones of each colour. A step consists of exchanging the position of two adjacent stones. Find the smallest positive integer

m

such that it is always possible to achieve, with at most

m

steps, that the

n

stones are together, if: a)

n

is even. b)

n

is odd and

k=3

2

Problem 2, Iberoamerican Olympiad 2011

Find all positive integers

n

for which exist three nonzero integers

x, y, z

such that

x+y+z=0

and:

\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=\frac{1}{n}

Problem 5, Iberoamerican Olympiad 2011

Let

x_1,\ldots ,x_n

be positive real numbers. Show that there exist

a_1,\ldots ,a_n\in\{-1,1\}

such that:

a_1x_1^2+a_2x_2^2+\ldots +a_nx_n^2\ge (a_1x_1+a_2x_2+\ldots + a_n x_n)^2

1

2

Problem 1, Iberoamerican Olympiad 2011

The number

2

is written on the board. Ana and Bruno play alternately. Ana begins. Each one, in their turn, replaces the number written by the one obtained by applying exactly one of these operations: multiply the number by

2

, multiply the number by

3

or add

1

to the number. The first player to get a number greater than or equal to

2011

wins. Find which of the two players has a winning strategy and describe it.

Problem 4, Iberoamerican Olympiad 2011

Let

ABC

be an acute-angled triangle, with

AC \neq BC

and let

O

be its circumcenter. Let

P

and

Q

be points such that

BOAP

and

COPQ

are parallelograms. Show that

Q

is the orthocenter of

ABC

.

2011 IberoAmerican

Subcontests

Problem 3, Iberoamerican Olympiad 2011

Problem 6, Iberoamerican Olympiad 2011

Problem 2, Iberoamerican Olympiad 2011

Problem 5, Iberoamerican Olympiad 2011

Problem 1, Iberoamerican Olympiad 2011

Problem 4, Iberoamerican Olympiad 2011