MathDB

p1. Two spiders mutually support each other only if the distance between them is greater than one meter. Determine the maximum number of spiders that can live on the web represented in the figure, where the square has a side of one meter and each extension measures

0.5

meters. https://cdn.artofproblemsolving.com/attachments/3/d/e32bd6c42e36e024b3a2cfb7bd3f57326e3950.jpg
p2. Investigate whether there exists a positive integer

N

, such that if the first digit of its decimal expression results in a number

p

such that

N

57

times

p

.
p3. A man, to hide his

2004

banknotes, distributes them among his books and then these books are kept in lockers in her library (a locker can hold multiple books, but it is not allows you to put a book inside another). It is known that in each locker in the library there are more books than tickets. Can one be certain that at least one of the books contains a quantity of banknotes that does not exceed the total number of lockers?
p4. Prove that for every positive integer

n

, the sum of the squares of all its positive integer divisors (including

1

and

n

) is different than

(n + 1)^2

.

p5. A tetrahedron in space is a set of

4

non-coplanar points (called vertices) joined in paire by

6

line segments (called edges). Prove that between the

4

vertices of each tetrahedron there is at least one with the property that the three edges that come out of this vertex can form the sides of a triangle (that is, the sum of the lengths of two of those edges is greater than the length of the third).
p6. Two people play "

2004

", a game that consists of making moves one at a time, starting with the number

2004

. The first player in his move must subtract from said number any of its integer divisors. The result then passes to his opponent who must also subtract from the result received in one of its integer divisors, and so on. The game is lost by the person who gets a

0

after completing his turn. Describe a strategy that enables whoever employs it to win always in this game.
p7. All the integer coordinate points

(m, n)

of the plane were painted in red colors, blue and green, so that all three colors are present in the resulting drawing. In addition, it is known that there are only

2004

points of a certain color (without knowing what this color is). Prove it exists by minus one right triangle angle in the plane whose vertices are integer coordinate points painted from different colors.

2004 Chile Classification NMO Juniors

Subcontests

2004 Chile Classification / Qualifying NMO Juniors XVI