MathDB
Problems
Contests
International Contests
Cono Sur Olympiad
2013 Cono Sur Olympiad
2013 Cono Sur Olympiad
Part of
Cono Sur Olympiad
Subcontests
(6)
6
1
Hide problems
Cono Sur Olympiad 2013, Problem 6
Let
A
B
C
D
ABCD
A
BC
D
be a convex quadrilateral. Let
n
≥
2
n \geq 2
n
≥
2
be a whole number. Prove that there are
n
n
n
triangles with the same area that satisfy all of the following properties:a) Their interiors are disjoint, that is, the triangles do not overlap. b) Each triangle lies either in
A
B
C
D
ABCD
A
BC
D
or inside of it. c) The sum of the areas of all of these triangles is at least
4
n
4
n
+
1
\frac{4n}{4n+1}
4
n
+
1
4
n
the area of
A
B
C
D
ABCD
A
BC
D
.
5
1
Hide problems
Cono Sur Olympiad 2013, Problem 5
Let
d
(
k
)
d(k)
d
(
k
)
be the number of positive divisors of integer
k
k
k
. A number
n
n
n
is called balanced if
d
(
n
−
1
)
≤
d
(
n
)
≤
d
(
n
+
1
)
d(n-1) \leq d(n) \leq d(n+1)
d
(
n
−
1
)
≤
d
(
n
)
≤
d
(
n
+
1
)
or
d
(
n
−
1
)
≥
d
(
n
)
≥
d
(
n
+
1
)
d(n-1) \geq d(n) \geq d(n+1)
d
(
n
−
1
)
≥
d
(
n
)
≥
d
(
n
+
1
)
. Show that there are infinitely many balanced numbers.
4
1
Hide problems
Cono Sur Olympiad 2013, Problem 4
Let
M
M
M
be the set of all integers from
1
1
1
to
2013
2013
2013
. Each subset of
M
M
M
is given one of
k
k
k
available colors, with the only condition that if the union of two different subsets
A
A
A
and
B
B
B
is
M
M
M
, then
A
A
A
and
B
B
B
are given different colors. What is the least possible value of
k
k
k
?
3
1
Hide problems
Cono Sur Olympiad 2013, Problem 3
Nocycleland is a country with
500
500
500
cities and
2013
2013
2013
two-way roads, each one of them connecting two cities. A city
A
A
A
neighbors
B
B
B
if there is one road that connects them, and a city
A
A
A
quasi-neighbors
B
B
B
if there is a city
C
C
C
such that
A
A
A
neighbors
C
C
C
and
C
C
C
neighbors
B
B
B
. It is known that in Nocycleland, there are no pair of cities connected directly with more than one road, and there are no four cities
A
A
A
,
B
B
B
,
C
C
C
and
D
D
D
such that
A
A
A
neighbors
B
B
B
,
B
B
B
neighbors
C
C
C
,
C
C
C
neighbors
D
D
D
, and
D
D
D
neighbors
A
A
A
. Show that there is at least one city that quasi-neighbors at least
57
57
57
other cities.
2
1
Hide problems
Cono Sur Olympiad 2013, Problem 2
In a triangle
A
B
C
ABC
A
BC
, let
M
M
M
be the midpoint of
B
C
BC
BC
and
I
I
I
the incenter of
A
B
C
ABC
A
BC
. If
I
M
IM
I
M
=
I
A
IA
I
A
, find the least possible measure of
∠
A
I
M
\angle{AIM}
∠
A
I
M
.
1
1
Hide problems
Cono Sur Olympiad 2013, Problem 1
Four distinct points are marked in a line. For each point, the sum of the distances from said point to the other three is calculated; getting in total 4 numbers.Decide whether these 4 numbers can be, in some order: a)
29
,
29
,
35
,
37
29,29,35,37
29
,
29
,
35
,
37
b)
28
,
29
,
35
,
37
28,29,35,37
28
,
29
,
35
,
37
c)
28
,
34
,
34
,
37
28,34,34,37
28
,
34
,
34
,
37