MathDB

1

2 player game a consecutive numbers with repetitions

a < b

, Xavier and Ze play the following game. First, Xavier writes

a

consecutive numbers of his choice; then, repeat some of them, also of his choice, until he has

b

numbers, with the condition that the sum of the

b

numbers written is an even number. Ze wins the game if he manages to separate the numbers into two groups with the same amount. Otherwise, Xavier wins. For example, for

a = 4

and

b = 7

, if Xavier wrote the numbers

3,4,5,6,3,3,4

, Ze could win, separating these numbers into groups

3,3 ,4,4

and

3,5,6

. For what values of

a

and

b

can Xavier guarantee victory?

5

1

10 professionals and 6 amateurs in a badminton competition

In a badminton competition,

16

players participate, of which

10

are professionals and

6

are amateurs. In the first phase, eight games are drawn. Among the eight winners of these games, four games are drawn. The four winners qualify for the semi-finals of the competition. Assuming that, whenever a professional player and an amateur play each other, the professional wins the game, what is the probability that an amateur player will reach the semi-finals of the competition?

3

1

partition in rectangles a 8x8 chessboard

The Proenc has a new

8\times 8

chess board and requires composing it into rectangles that do not overlap, so that: (i) each rectangle has as many white squares as black ones; (ii) there are no two rectangles with the same number of squares. Determines the maximum value of

n

for which such a decomposition is possible. For this value of

n

, determine all possible sets

{A_1,... ,A_n}

, where

A_i

is the number of rectangle

i

in squares, for which a decomposition of the board under the conditions intended actions is possible.

1

15 students, with different heights

Raul's class has

15

students, all with different heights. The Mathematics teacher wants to place them in a queue so that, at the beginning of the queue, they are ordered in ascending order of heights, from then on, they are ordered in descending order and Raul, who He is the tallest in the class, he cannot be at the extremes. In how many different ways is it possible to form this queue?

2

1

tangent circles wanted

Let

P

be a point on a circle

C_1

and let

C_2

be a circle with center

P

that intersects

C_1

at two points Q and R. The circle

C_3

, with center

Q

and which passes through

R

, intersects

C_2

at another point S, as in figure. Shows that

QS

is tangent to

C_1

. https://cdn.artofproblemsolving.com/attachments/7/5/f48d414c68c33c4efaf4d6c8bebcf6f1fad4ba.png

4

1

< BAD = 30^o if <ADB = 45^o, <ACB = 30^o for median AD of ABC

Let

[AD]

be a median of the triangle

[ABC]

. Knowing that

\angle ADB = 45^o

and

\angle A CB = 30^o

, prove that

\angle BAD = 30^o

.

2022 Portugal MO

Subcontests

2 player game a consecutive numbers with repetitions

10 professionals and 6 amateurs in a badminton competition

partition in rectangles a 8x8 chessboard

15 students, with different heights

tangent circles wanted

< BAD = 30^o if <ADB = 45^o, <ACB = 30^o for median AD of ABC

2022 Portugal MO

Subcontests

2 player game a consecutive numbers with repetitions

10 professionals and 6 amateurs in a badminton competition

partition in rectangles a 8x8 chessboard

15 students, with different heights

tangent circles wanted

&lt; BAD = 30^o if &lt;ADB = 45^o, &lt;ACB = 30^o for median AD of ABC

< BAD = 30^o if <ADB = 45^o, <ACB = 30^o for median AD of ABC