MathDB

[url=https://artofproblemsolving.com/community/c1068820h2935473p26269278]p1. Find all the solutions

(a, b, c)

in the natural numbers, verifying

1\le a \le b \le c

, of the equation

\frac34=\frac{1}{a}+\frac{1}{b}+\frac{1}{c}

[url=https://artofproblemsolving.com/community/c4h1845726p12426532]p2. Inside a cube of side

1

, two spheres are introduced that are tangent externally to each other and such that each is tangent to three faces of the cube. Determine the greater distance to which the centers of the spheres can be found.
[url=https://artofproblemsolving.com/community/c1068820h2935475p26269289]p3. Consider the following figure formed by

10

nodes and

15

edges: https://cdn.artofproblemsolving.com/attachments/8/e/205cf553f6d7d8658e19c3ff522dac085761a9.png Prove that the edges of the figure cannot be colored by using

3

different colors so that the edges that reach each node have different colors from each other.
p4. In each square of a

9\times 9

squared board, write a natural number, in such a way that the difference of the numbers located in adjacent squares is at most

3

. Show that there are at least two boxes that have the same number written on them.
Clarification: two squares are adjacent when they have a side in common.
PS. Problems 1 and 3 were also proposed as Seniors P1 and P3.

2011 Chile Junior Math Olympiad

Subcontests

2011 Chile NMO Juniors XXIII