2002 Portugal MO

Part of Portugal MO

Subcontests

(6)

1

10 pople in a a bookstore for science.

6

2002

, the celebrations of the

500

th anniversary of the birth of by mathematician Pedro Nunes. That morning, only ten people entered the Viva bookstore for science. Each of these people bought exactly

3

different books. Furthermore, any two people bought at least one copy of the same book. The Adventures of Mathematics by Pedro Nunes was one of the books that achieved the highest number of sales in this morning. What is the smallest value this number could have taken?

1

no 2 Blablabla numbers exist such one of them divides the other.

The Blablabla set contains all the different seven-digit numbers that can be formed with the digits

2, 3, 4, 5, 6, 7

8

. Prove that there are not two Blablabla numbers such that one of them is divisible by the other.

1

2 points with same colore at distance \sqrt5 , 4 colours in 2x4

Daniel painted a rectangular painting measuring

2

4

meters with four colors. Knowing that he used more than two colors to paint the four corners of the painting, prove that he painted of the same color two points that are at least

\sqrt5

1

881st position of different words alphabetically7 letters A, N, A, V, I, S, O

The keyword that Ana Viso chose for her computer has the

7

characters of her name: A, N, A, V, I, S, O. Sorting all the different words alphabetically formed by all these

7

characters, Ana's keyword appears in the

881

st position. What it is Ana's keyword?

1

2 squares, integer sides, perfect square sum of areas 2002 Portugal p5

Consider the three squares indicated in the figure. Show that if the lengths of the sides of the smaller square and the square greater are integers, then adding to the area of the smallest square the area of the inclined square, a perfect square is obtained. https://1.bp.blogspot.com/-B0QdvZIjOLw/X4URvs3C0ZI/AAAAAAAAMmw/S5zMpPBXBn8Jj39d-OZVtMRUDn3tXbyWgCLcBGAsYHQ/s0/2002%2Bportugal%2Bp5.png

1

computational, 5 spheres, 1 on top of other 4, 2002 Portugal p2

Consider five spheres with radius

10

cm . Four of these spheres are arranged on a horizontal table so that its centers form a

20

cm square and the fifth sphere is placed on them so that it touches the other four. What is the distance between center of this fifth sphere and the table?