MathDB

1

FINALS - Portuguese Mathematical Olympiad (High-School) P6

One hundred musicians are planning to organize a festival with several concerts. In each concert, while some of the one hundred musicians play on stage, the others remain in the audience assisting to the players. What is the least number of concerts so that each of the musicians has the chance to listen to each and every one of the other musicians on stage?

5

1

FINALS - Portuguese Mathematical Olympiad (High-School) P5

Let

[ABCD]

be a convex quadrilateral with area

2014

, and let

P

be a point on

[AB]

and

Q

a point on

[AD]

such that triangles

[ABQ]

and

[ADP]

have area

1

. Let

R

be the intersection of

[AC]

and

[PQ]

. Determine

\frac{\overline{RC}}{\overline{RA}}

.

4

1

FINALS - Portuguese Mathematical Olympiad (High-School) P4

Determine all natural numbers

x

,

y

and

z

, such that

x\leq y\leq z

and

\left(1+\frac1x\right)\left(1+\frac1y\right)\left(1+\frac1z\right) = 3\text{.}

3

1

FINALS - Portuguese Mathematical Olympiad (High-School) P3

Amélia and Beatriz play battleship on a

2n\times2n

board, using very peculiar rules. Amélia begins by choosing

n

lines and

n

columns of the board, placing her

n^2

submarines on the cells that lie on their intersections. Next, Beatriz chooses a set of cells that will explode. Which is the least number of cells that Beatriz has to choose in order to assure that at least a submarine will explode?

2

1

FINALS - Portuguese Mathematical Olympiad (High-School) P2

Let

[ABCD]

be a square,

M

a point on the segment

[AD]

, and

N

a point on the segment

[DC]

such that

B\hat{M}A = N\hat{M}D = 60^{\circ}

. Calculate

M\hat{B}N

.

1

FINALS - Portuguese Mathematical Olympiad (High-School) P1

The ship Meridiano do Bacalhau does its fishing business during

64

days. Each day the capitain chooses a direction which may be either north or south and the ship sails that direction in that day. On the first day of business the ship sails

1

mile, on the second day sails

2

miles; generally, on the

n

-th day it sails

n

miles. After of the

64

-th day, the ship was

2014

miles north from its initial position. What is the greatest number of days that the ship could have sailed south?

2014 Portugal MO

Subcontests

FINALS - Portuguese Mathematical Olympiad (High-School) P6

FINALS - Portuguese Mathematical Olympiad (High-School) P5

FINALS - Portuguese Mathematical Olympiad (High-School) P4

FINALS - Portuguese Mathematical Olympiad (High-School) P3

FINALS - Portuguese Mathematical Olympiad (High-School) P2

FINALS - Portuguese Mathematical Olympiad (High-School) P1