MathDB

1

there is always a multiple of n that can be written only with digits 0 and 1

n

be a natural number. Prove that there is a multiple of

n

that can be written only with the digits

0

and

1

.

4

1

in each row and column of chessboard there is an odd number of pieces

During a game of chess, at a certain point, in each row and column of the board there is an odd number of pieces. Prove that the number of pieces that are on black squares is even. (Note: a chessboard has

8

rows and

8

columns)

3

1

consecutive zeros at the end of the 2001!

How many consecutive zeros are there at the end of the number

2001! = 2001 \times 2000 \times ... \times 3 \times 2 \times 1

?

1

4 teams by 7 members

To be able to play the Game of Glory, seven friends have to form four teams which, obviously, may not have the same number of members. In how many different ways can they form these teams? (The order of people within teams is not important, but each person can only be part of one team)

5

1

computational, 6 equal spheres around a cone 2001 Portugal p5

On a table are a cone, resting on the base, and six equal spheres tangent to the cone. Besides that, each sphere is tangent to the two adjacent spheres. Knowing that the radius

R

of the base of the cone is half its height and determine the radius

r

of the spheres.

2

1

computational in trapezoid, BC = 2 DA, <DAB + <ABC =120^o 2001 Portugal p2

The trapezium

[ABCD]

has bases

[AB]

and

[CD]

(with

[AB]

being the largest base). Knowing that

BC = 2 DA

and that

\angle DAB + \angle ABC =120^o

, determines the measure of

\angle DAB

.

2001 Portugal MO

Subcontests

there is always a multiple of n that can be written only with digits 0 and 1

in each row and column of chessboard there is an odd number of pieces

consecutive zeros at the end of the 2001!

4 teams by 7 members

computational, 6 equal spheres around a cone 2001 Portugal p5

computational in trapezoid, BC = 2 DA, <DAB + <ABC =120^o 2001 Portugal p2

2001 Portugal MO

Subcontests

there is always a multiple of n that can be written only with digits 0 and 1

in each row and column of chessboard there is an odd number of pieces

consecutive zeros at the end of the 2001!

4 teams by 7 members

computational, 6 equal spheres around a cone 2001 Portugal p5

computational in trapezoid, BC = 2 DA, &lt;DAB + &lt;ABC =120^o 2001 Portugal p2

computational in trapezoid, BC = 2 DA, <DAB + <ABC =120^o 2001 Portugal p2