MathDB

1

3-person competition

Rosa dos Ventos, Aurora Boreal and Manuela do Norte organized a competition between them last weekend, consisting of several athletics events. The winner in each test obtained

x

points, the second placed

y

points and the third placed

z

points (

x,y,z \in N

and

x >y>z

). The final result of the competition, obtained by adding up the scores in each event, was Rosa had

22

points, Manuela had

9

points, Aurora had

9

points. In how many tests did they compete and who came second in the high jump knowing that the Manuela won the

100

meters and no one gave up in any race?
[hide=official wording]Rosa dos Ventos, a Aurora Boreal e a Manuela do Norte organizaram no passado fim de semana uma competi¸c˜ao entre elas, consistindo em v´arias provas de atletismo. A vencedora em cada prova obteve x pontos, a segunda classificada y pontos e a terceira classificada z pontos (x,y,z ∈ IN e x >y>z). O resultado final da competi¸c˜ao, obtido por soma das pontua¸c˜oes em cada prova, foi Rosa 22 pontos Manuela 9 pontos Aurora 9 pontos Em quantas provas competiram e quem ficou em segundo lugar no salto em altura sabendo que a Manuela ganhou os 100 metros e que ningu´em desistiu em nenhuma prova?

6

1

x is rational iff x, x+1, ..., x+n, .. contains at least 3 terms in geom. progr

Prove that a real number

x

is rational if and only if the sequence

x, x+1, x+2, x+3, ..., x+n, ...

contains, at least least three terms in geometric progression.

2

1

arrest 2 of 5 criminals to lacate the leader

Through an informant, the police know the meeting place of a group of criminals. The identity of the different elements of the group is, however, unknown. Inspector Loureiro's task is to arrest the leader of the group. The inspector knows that the leader of the group is the shortest of the five members of the group, all of them of different heights, who will be present at the meeting. After the meeting, the bandits - as a precautionary measure - leave the building separately with an interval of

15

minutes. As the inspector doesn't know which of them is the shortest, he decides to let the first two criminals out, and arrest the first of the following who is shorter than those who left until that moment. What is the probability that Inspector Loureiro will arrest the right person?

1

Joao jumps 2^{k-1} meters at jump k

Joao Salta-Pocinhas jumps

1

meter in the first jump,

2

meters in the second,

4

meters in the third, . . .,

2^{n-1}

meters in jump number

n

. Is there any possibility for Joao to choose the directions of his jumps in order to get back to the starting point?

4

1

area wanted, secants in a circle 1995 Portugal p4

The diameter

[AC]

of a circle is divided into four equal segments by points

P, M

and

Q

. Consider a segment

[BD]

that passes through

P

and cuts the circle at

B

and

D

, such that

PD =\frac{3}{2} AP

. Knowing that the area of the triangle

[ABP]

has measure

1

cm

^2

, calculate the area of

[ABCD]

? https://1.bp.blogspot.com/-ibre0taeRo8/X4KiWWSROEI/AAAAAAAAMl4/xFNfpQBxmMMVLngp5OWOXRLMuaxf3nolQCLcBGAsYHQ/s154/1995%2Bportugal%2Bp5.png

3

1

3 ants moving, each at line // of other 2, rectangle 1995 Portugal p3

Three ants are at three corners of a rectangle. It is assumed that each ant moves only when the other two are stopped and always parallel to the line defined by them. Will be is it possible that the three ants are simultaneously at midpoints on the sides of the rectangle?

1995 Portugal MO

Subcontests

3-person competition

x is rational iff x, x+1, ..., x+n, .. contains at least 3 terms in geom. progr

arrest 2 of 5 criminals to lacate the leader

Joao jumps 2^{k-1} meters at jump k

area wanted, secants in a circle 1995 Portugal p4

3 ants moving, each at line // of other 2, rectangle 1995 Portugal p3