MathDB

2

Giant square - Portuguese MO, Problem 3, 2009

Duarte wants to draw a square whose side's length is

2009

cm and which is divided in

2009\times2009

squares whose side's length is

1

cm and whose sides are parallel to the original square's one, without taking the pencil out of the paper. Starting on one of the vertex of the giant square, what is the length of the shortest line that allows him to make this drawing?

Explorers, Traps and Stones - Portuguese MO, Problem 6, 2009

Two players play the following game on a circular board with 2009 houses. The two plays put, alternatively, on an empty house, one of three pieces, called explorer (E), trap (T) or stone (S). A treasure is a sequence of three consecutive filled houses such that the first one (on any direction) has an explorer and the middle one doesn't have a trap. For example, STE is not a treasure, while TEE is a treasure. The first player forming a treasure wins. Can any of the players guarantee the victory? And, in affirmative case, who?

2

South African converse - Portuguese MO, Problem 2, 2009

Points

N

and

M

are on the sides

CD

and

BC

of square

ABCD

, respectively. The perimeter of triangle

MCN

is equal to the double of the length of the square's side. Find

\angle MAN

.

Hong Kong easy part - Portuguese MO, Problem 5, 2009

Circumferences

C_1

and

C_2

have different radios and are externally tangent on point

T

. Consider points

A

on

C_1

and

B

on

C_2

, both different from

T

, such that

\angle BTA=90^{\circ}

. What is the locus of the midpoints of line segments

AB

constructed that way?

1

2

Circumference invariance - Portuguese MO, Problem 1, 2009

A circumference was divided in

n

equal parts. On each of these parts one number from

1

to

n

was placed such that the distance between consecutive numbers is always the same. Numbers

11

,

4

and

17

were in consecutive positions. In how many parts was the circumference divided?

Giant sum of products - Portuguese MO, Problem 4, 2009

João calculated the product of the non zero digits of each integer from

1

to

10^{2009}

and then he summed these

10^{2009}

products. Which number did he obtain?

2009 Portugal MO

Subcontests

Giant square - Portuguese MO, Problem 3, 2009

Explorers, Traps and Stones - Portuguese MO, Problem 6, 2009

South African converse - Portuguese MO, Problem 2, 2009

Hong Kong easy part - Portuguese MO, Problem 5, 2009

Circumference invariance - Portuguese MO, Problem 1, 2009

Giant sum of products - Portuguese MO, Problem 4, 2009