2011 Chile National Olympiad

Part of Chile National Olympiad

Subcontests

(4)

1

map with 30 different cities

It is intended to make a map locating

30

different cities on it. For this, all the distances between these cities are available as data (each of these distances is considered as a “data”). Three of these cities are already laid out on the map, and they turn out to be non-collinear. How much data must be used as a minimum to complete the map?

1

3 colors for a figure by 10 nodes and 15 edges

Consider the following figure formed by

10

15

edges:
[asy] unitsize(1.5 cm);
pair A, B, C, D, E, F, G, H, I, J;
A = dir(90); B = dir(90 + 360/5); C = dir(90 + 2*360/5); D = dir(90 + 3*360/5); E = dir(90 + 4*360/5); F = 0.6*A; G = 0.6*B; H = 0.6*C; I = 0.6*D; J = 0.6*E;
draw(A--B--C--D--E--cycle); draw(F--H--J--G--I--cycle); draw(A--F); draw(B--G); draw(C--H); draw(D--I); draw(E--J);
dot(A); dot(B); dot(C); dot(D); dot(E); dot(F); dot(G); dot(H); dot(I); dot(J); [/asy]
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.

1

3/4 =1/a+1/b+1/c diophantine

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}.

1

sum of 3 angles of 3 pairs of tangents of 3 circles is 180^o Chile 2011 L2 P2

O

be the center of the circle circumscribed to triangle

ABC

S_ {A}

S_ {B}

S_ {C}

be the circles centered on

O

that are tangent to the sides

BC, CA, AB

respectively. Show that the sum of the angle between the two tangents

S_ {A}

A

plus the angle between the two tangents

S_ {B}

B

plus the angle between the two tangents

S_ {C}

C

180