MathDB

1

no numbers in 6 chosen from a total of 36 numbers

1

to

36

are written in each "Neuro-Millions" bulletin. A bet on "Neuro-Millions" consists of choosing

6

of these

36

numbers. Then,

6

numbers between

1

and

36

are drawn, and these constitute the key to "Neuro-Milh˜oes". A bet is awarded if it does not contain any of the key numbers. How many bets, at least, are necessary to guarantee a prize?

5

1

exactly one fifth of the natural numbers 1,2,...,n are divisors of n.

Determine all the natural numbers

n

such that exactly one fifth of the natural numbers

1,2,...,n

are divisors of

n

.

3

1

n bags with 1 bean, n bags with 3 beans, ..., n bags with 2006 beans

A bean packing plant has a machine that puts a certain amount of beans into bags and then puts a certain amount of bags into boxes, which are then shipped to customers. One day, the machine broke down and the first n bags came out empty, the next

n

bags came out with

1

bean, the next

n

bags came out with

2

beans,..., and the last

n

bags came out with

2006

beans. To provide each customer with the agreed quantity of bags of beans, the person responsible for the unit intends to distribute the bags among the

2007

boxes that day so that all boxes contain the same number of bags and all boxes contain the same number. number of beans. For what values of

n

is this possible?

1

two men wlak as train passes

Alexandre and Herculano are at Campanha station waiting for the train. To entertain themselves, they decide to calculate the length of a freight train that passes through the station without changing its speed. When the front of the train passes them, Alexandre starts walking in the direction of the train's movement and Herculano starts walking in the opposite direction. The two walk at the same speed and each of them stops at the moment they cross the end of the train. Alexandre walked

45

meters and Herculano

30

. How long is the train?

4

1

isosceles inside a parallelogram wanted, midpoint, projection 2006 Portugal p4

In the parallelogram

[ABCD], E

is the midpoint of

[AD]

and

F

the orthogonal projection of

B

on

[CE]

. Prove that the triangle

[ABF]

is isosceles. https://1.bp.blogspot.com/-DLmFg8ayEQ4/X4XMohA5TjI/AAAAAAAAMnk/thlIKnNUiCkuu9cg1Aq7Zltz8SenmFWuwCLcBGAsYHQ/s0/2006%2Bportugal%2Bp4.png

2

1

perpendicular wanted, inside an equilateral 2006 Portugal p2

In the equilateral triangle

[ABC], D

is the midpoint of

[AC], E

and the orthogonal projection of

D

over

[CB]

and

F

is the midpoint of

[DE]

. Prove that

[FB]

and

[AE]

are perpendicular. https://1.bp.blogspot.com/-TjSyQotGIOM/X4XMolaXHvI/AAAAAAAAMng/cVsHfl-lrXAFE5LMdosE6vqK1Tf-8WOQgCLcBGAsYHQ/s0/2006%2Bportugal%2Bp2.png

2006 Portugal MO

Subcontests

no numbers in 6 chosen from a total of 36 numbers

exactly one fifth of the natural numbers 1,2,...,n are divisors of n.

n bags with 1 bean, n bags with 3 beans, ..., n bags with 2006 beans

two men wlak as train passes

isosceles inside a parallelogram wanted, midpoint, projection 2006 Portugal p4

perpendicular wanted, inside an equilateral 2006 Portugal p2