MathDB

2

Problem 3, Iberoamerican Olympiad 2010

\Gamma

is inscribed to the scalene triangle

ABC

.

\Gamma

is tangent to the sides

BC, CA

and

AB

at

D, E

and

F

respectively. The line

EF

intersects the line

BC

at

G

. The circle of diameter

GD

intersects

\Gamma

in

R

(

R\neq D

). Let

P

,

Q

(

P\neq R , Q\neq R

) be the intersections of

\Gamma

with

BR

and

CR

, respectively. The lines

BQ

and

CP

intersects at

X

. The circumcircle of

CDE

meets

QR

at

M

, and the circumcircle of

BDF

meet

PR

at

N

. Prove that

PM

,

QN

and

RX

are concurrent.
Author: Arnoldo Aguilar, El Salvador

Problem 6, Iberoamerican Olympiad 2010

Around a circular table sit

12

people, and on the table there are

28

vases. Two people can see each other, if and only if there is no vase lined with them. Prove that there are at least two people who can be seen.

2

Problem 2, Iberoamerican Olympiad 2010

Determine if there are positive integers

a, b

such that all terms of the sequence defined by x_{1}= 2010,x_{2}= 2011\\ x_{n+2}= x_{n}+ x_{n+1}+a\sqrt{x_{n}x_{n+1}+b} (n\ge 1) are integers.

Problem 5, Iberoamerican Olympiad 2010

Let

ABCD

be a cyclic quadrilateral whose diagonals

AC

and

BD

are perpendicular. Let

O

be the circumcenter of

ABCD

,

K

the intersection of the diagonals,

L\neq O

the intersection of the circles circumscribed to

OAC

and

OBD

, and

G

the intersection of the diagonals of the quadrilateral whose vertices are the midpoints of the sides of

ABCD

. Prove that

O, K, L

and

G

are collinear

1

2

Problem 1, Iberoamerican Olympiad 2010

There are ten coins a line, which are indistinguishable. It is known that two of them are false and have consecutive positions on the line. For each set of positions, you may ask how many false coins it contains. Is it possible to identify the false coins by making only two of those questions, without knowing the answer to the first question before making the second?

Problem 4, Iberoamerican Olympiad 2010

The arithmetic, geometric and harmonic mean of two distinct positive integers are different numbers. Find the smallest possible value for the arithmetic mean.

2010 IberoAmerican

Subcontests

Problem 3, Iberoamerican Olympiad 2010

Problem 6, Iberoamerican Olympiad 2010

Problem 2, Iberoamerican Olympiad 2010

Problem 5, Iberoamerican Olympiad 2010

Problem 1, Iberoamerican Olympiad 2010

Problem 4, Iberoamerican Olympiad 2010