2018 Portugal MO

Part of Portugal MO

Subcontests

(6)

1

ten numbers at 10 vertices of 9-triangle grid

A triangle is divided into nine smaller triangles, where counters with the number zero are placed at each of the ten vertices. A movement consists of choosing one of the nine triangles and applying the same operation to the three counters of that triangle: adding a unit or subtracting a unit. The figure illustrates a possible movement. We shall call the integer number n dominant if it is possible, after a few moves, to obtain a configuration in which the counter numbers are all consecutive and the largest of these numbers is

n

. Determine all dominant numbers. https://cdn.artofproblemsolving.com/attachments/7/3/731160e6e9a2b3292a31c4555d4adbc7028596.png

1

guards in a museum

A museum wants to protect its most valuable piece by maintaining constant surveillance. To do this, he wants to place guards to watch the place, in shifts of

7

consecutive hours. Each guard starts his shift at the same time every day. A guard is essential if there is any time during the day when you are alone to watch the item. Indicates all possibilities for the number of guards guarding the piece, so that everyone is indispensable.

1

4 color of m x n board

How many ways are there to paint an

m \times n

board, so that each square is painted blue, white, brown or gold, and in each

2 \times 2

square there is one square of each color?

1

sum of 3 smallest divisors is 17, of max 3905 - Portugal OPM 2018 p1

Arthur thought of a positive integer and noticed that the sum of its three smallest divisors is

17

and that the sum of its three largest divisors is

3905

. Indicate all the numbers that Arthur may have thought of.

1

square interior to a square, 5 equal incircles, 4 right triangles

[ABCD]

is a square of side

1

E, F, G

H

[AFB], [BGC], [CHD]

[DEA]

are right-angled triangles. Knowing that the circles inscribed in each of these triangles and the circle inscribed in the square

[EFGH]

has all the same radius, what is the measure of the radius of the circles? https://1.bp.blogspot.com/-l37AEXa7_-c/X4KaJwe6HQI/AAAAAAAAMk4/14wvIipf26cRge_GqKSRwH32bp291vX4QCLcBGAsYHQ/s0/2018%2Bportugal%2Bp2.png

1

symmetrics of circumcenter wrt sides create a congruent triangle

[ABC]

be any triangle and let

D, E

F

be the symmetrics of the circumcenter wrt the three sides. Prove that the triangles

[ABC]

[DEF]

are congruent. https://cdn.artofproblemsolving.com/attachments/c/6/45bd929dfff87fb8deb09eddb59ef46e0dc0f4.png