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International Contests
Rioplatense Mathematical Olympiad, Level 3
2011 Rioplatense Mathematical Olympiad, Level 3
2011 Rioplatense Mathematical Olympiad, Level 3
Part of
Rioplatense Mathematical Olympiad, Level 3
Subcontests
(6)
5
1
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maximum number of ''angles'' of size 11 in a form of size 400
A form is the union of squared rectangles whose bases are consecutive unitary segments in a horizontal line that leaves all the rectangles on the same side, and whose heights
m
1
,
.
.
.
,
m
n
m_1, ... , m_n
m
1
,
...
,
m
n
satisying
m
1
≥
.
.
.
≥
m
n
m_1\ge ... \ge m_n
m
1
≥
...
≥
m
n
. An angle in a form consists of a box
v
v
v
and of all the boxes to the right of
v
v
v
and all the boxes above
v
v
v
. The size of a form of an angle is the number of boxes it contains. Find the maximum number of angles of size
11
11
11
in a form of size
400
400
400
.[url=http://www.oma.org.ar/enunciados/omr20.htm]source
4
1
Hide problems
parallelogram from intersections, starting with 2 intersecting circles
We consider
Γ
1
\Gamma_1
Γ
1
and
Γ
2
\Gamma_2
Γ
2
two circles that intersect at points
P
P
P
and
Q
Q
Q
. Let
A
,
B
A , B
A
,
B
and
C
C
C
be points on the circle
Γ
1
\Gamma_1
Γ
1
and
D
,
E
D , E
D
,
E
and
F
F
F
points on the circle
Γ
2
\Gamma_2
Γ
2
so that the lines
A
E
A E
A
E
and
B
D
B D
B
D
intersect at
P
P
P
and the lines
A
F
A F
A
F
and
C
D
C D
C
D
intersect at
Q
Q
Q
. Denote
M
M
M
and
N
N
N
the intersections of lines
A
B
A B
A
B
and
D
E
D E
D
E
and of lines
A
C
A C
A
C
and
D
F
D F
D
F
, respectively. Show that
A
M
D
N
A M D N
A
M
D
N
is a parallelogram.
3
1
Hide problems
map of flights with routes, coloring related
Let
M
M
M
be a map made of several cities linked to each other by flights. We say that there is a route between two cities if there is a nonstop flight linking these two cities. For each city to the
M
M
M
denote by
M
a
M_a
M
a
the map formed by the cities that have a route to and routes linking these cities together ( to not part of
M
a
M_a
M
a
). The cities of
M
a
M_a
M
a
are divided into two sets so that the number of routes linking cities of different sets is maximum; we call this number the cut of
M
a
M_a
M
a
. Suppose that for every cut of
M
a
M_a
M
a
, it is strictly less than two thirds of the number of routes
M
a
M_a
M
a
. Show that for any coloring of the
M
M
M
routes with two colors there are three cities of
M
M
M
joined by three routes of the same color.
6
1
Hide problems
Divisors and Euler's Theorem
Let
d
(
n
)
d(n)
d
(
n
)
be the sum of positive integers divisors of number
n
n
n
and
ϕ
(
n
)
\phi(n)
ϕ
(
n
)
the quantity of integers in the interval
[
0
,
n
]
[0,n]
[
0
,
n
]
such that these integers are coprime with
n
n
n
. For instance
d
(
6
)
=
12
d(6)=12
d
(
6
)
=
12
and
ϕ
(
7
)
=
6
\phi(7)=6
ϕ
(
7
)
=
6
. Determine if the set of the integers
n
n
n
such that,
d
(
n
)
⋅
ϕ
(
n
)
d(n)\cdot \phi (n)
d
(
n
)
⋅
ϕ
(
n
)
is a perfect square, is finite or infinite set.
2
1
Hide problems
Triangle similarity
Let
A
B
C
ABC
A
BC
an acute triangle and
H
H
H
its orthocenter. Let
E
E
E
and
F
F
F
be the intersection of lines
B
H
BH
B
H
and
C
H
CH
C
H
with
A
C
AC
A
C
and
A
B
AB
A
B
respectively, and let
D
D
D
be the intersection of lines
E
F
EF
EF
and
B
C
BC
BC
. Let
Γ
1
\Gamma_1
Γ
1
be the circumcircle of
A
E
F
AEF
A
EF
, and
Γ
2
\Gamma_2
Γ
2
the circumcircle of
B
H
C
BHC
B
H
C
. The line
A
D
AD
A
D
intersects
Γ
1
\Gamma_1
Γ
1
at point
I
≠
A
I \neq A
I
=
A
. Let
J
J
J
be the feet of the internal bisector of
∠
B
H
C
\angle{BHC}
∠
B
H
C
and
M
M
M
the midpoint of the arc
B
C
⌢
\stackrel{\frown}{BC}
BC
⌢
from
Γ
2
\Gamma_2
Γ
2
that contains the point
H
H
H
. The line
M
J
MJ
M
J
intersects
Γ
2
\Gamma_2
Γ
2
at point
N
≠
M
N \neq M
N
=
M
. Show that the triangles
E
I
F
EIF
E
I
F
and
C
N
B
CNB
CNB
are similar.
1
1
Hide problems
Number followers
Given a positive integer
n
n
n
, an operation consists of replacing
n
n
n
with either
2
n
−
1
2n-1
2
n
−
1
,
3
n
−
2
3n-2
3
n
−
2
or
5
n
−
4
5n-4
5
n
−
4
. A number
b
b
b
is said to be a follower of number
a
a
a
if
b
b
b
can be obtained from
a
a
a
using this operation multiple times. Find all positive integers
a
<
2011
a < 2011
a
<
2011
that have a common follower with
2011
2011
2011
.