MathDB

p1. Consider a number of two or more digits, none of them being zero

(0)

. Such a number is called a thirteen if every pair of consecutive digits forms a number divisible by

13

. For example:

139

is thirteen since

13 = 13\times 1

and

39 = 13\times 3

. How many thirteen five-digit numbers are there?
p2. Katia and Mariela play the following game: In each of their three turns, Katia replaces one of the stars in the expression

\star \star\star\star\star\star\star\star\star

for some a digit of between

1

2

3

4

5

6

7

8

9

that has not been used before in the game. In shifts of Mariela, she replaces two of the stars with two different digits that have not been used. Katia starts playing and plays alternately. Mariela wins if the resulting number at the end of the game is divisible by

27

. Does Mariela have any to ensure the triumph?

p3. A rectangle has been divided, using three segments, into four sections, as illustrated in the figure. If the areas of the indicated sectors are

3

4

and

5

respectively, find the area of the sector in gray. https://cdn.artofproblemsolving.com/attachments/a/8/1e0ce85a694eb32a4bc0f42a8b6c38e071c38b.png
p4. On the surface of a cylinder of height

12

meters and base circle of perimeter

4

meters, a rope is wound whose initial end is on the circle upper, makes a total of four full turns of the cylinder and its lower end is at the lower circle. What is the shortest length that this string can have?
PS. Juniors P2 was also [url=https://artofproblemsolving.com/community/c4h2690902p23356652] Seniors P1

2016 Chile Classification NMO Juniors

Subcontests

2016 Chile Classification / Qualifying NMO Juniors XXVIII