MathDB

2

ABC

be an acute-angled triangle and let

AN

,

BM

and

CP

the altitudes with respect to the sides

BC

,

CA

and

AB

, respectively. Let

R

,

S

be the pojections of

N

on the sides

AB

,

CA

, respectively, and let

Q

,

W

be the projections of

N

on the altitudes

BM

and

CP

, respectively.
(a) Show that

R

,

Q

,

W

,

S

are collinear. (b) Show that

MP=RS-QW

.

Cono Sur Olympiad 2005, Problem 5

We say that a number of 20 digits is special if its impossible to represent it as an product of a number of 10 digits by a number of 11 digits. Find the maximum quantity of consecutive numbers that are specials.

1

2

Cono Sur Olympiad 2005, Problem 1

Let

a_n

be the last digit of the sum of the digits of

20052005...2005

, where the

2005

block occurs

n

times. Find

a_1 +a_2 + \dots +a_{2005}

.

Cono Sur Olympiad 2005, Problem 4

Let

ABC

be a isosceles triangle, with

AB=AC

. A line

r

that pass through the incenter

I

of

ABC

touches the sides

AB

and

AC

at the points

D

and

E

, respectively. Let

F

and

G

be points on

BC

such that

BF=CE

and

CG=BD

. Show that the angle

\angle FIG

is constant when we vary the line

r

.

2005 Cono Sur Olympiad

Subcontests

Cono Sur Olympiad 2005, Problem 3

Cono Sur Olympiad 2005, Problem 6

Cono Sur Olympiad 2005, Problem 2

Cono Sur Olympiad 2005, Problem 5

Cono Sur Olympiad 2005, Problem 1

Cono Sur Olympiad 2005, Problem 4