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Rioplatense Mathematical Olympiad, Level 3
2013 Rioplatense Mathematical Olympiad, Level 3
2013 Rioplatense Mathematical Olympiad, Level 3
Part of
Rioplatense Mathematical Olympiad, Level 3
Subcontests
(6)
6
1
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4 points lie on a circumference
Let
A
B
C
ABC
A
BC
be an acute scalene triangle,
H
H
H
its orthocenter and
G
G
G
its geocenter. The circumference with diameter
A
H
AH
A
H
cuts the circumcircle of
B
H
C
BHC
B
H
C
in
A
′
A'
A
′
(
A
′
≠
H
A' \neq H
A
′
=
H
). Points
B
′
B'
B
′
and
C
′
C'
C
′
are defined similarly. Show that the points
A
′
A'
A
′
,
B
′
B'
B
′
,
C
′
C'
C
′
, and
G
G
G
lie in one circumference.
4
1
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Playing on a convex n-agon and winning strategies
Two players
A
A
A
and
B
B
B
play alternatively in a convex polygon with
n
≥
5
n \geq 5
n
≥
5
sides. In each turn, the corresponding player has to draw a diagonal that does not cut inside the polygon previously drawn diagonals. A player loses if after his turn, one quadrilateral is formed such that its two diagonals are not drawn.
A
A
A
starts the game. For each positive integer
n
n
n
, find a winning strategy for one of the players.
3
1
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Partitions of groups of people
A division of a group of people into various groups is called
k
k
k
-regular if the number of groups is less or equal to
k
k
k
and two people that know each other are in different groups. Let
A
A
A
,
B
B
B
, and
C
C
C
groups of people such that there are is no person in
A
A
A
and no person in
B
B
B
that know each other. Suppose that the group
A
∪
C
A \cup C
A
∪
C
has an
a
a
a
-regular division and the group
B
∪
C
B \cup C
B
∪
C
has a
b
b
b
-regular division. For each
a
a
a
and
b
b
b
, determine the least possible value of
k
k
k
for which it is guaranteed that the group
A
∪
B
∪
C
A \cup B \cup C
A
∪
B
∪
C
has a
k
k
k
-regular division.
2
1
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Prove that two lines are perpendicular
Let
A
B
C
D
ABCD
A
BC
D
be a square, and let
E
E
E
and
F
F
F
be points in
A
B
AB
A
B
and
B
C
BC
BC
respectively such that
B
E
=
B
F
BE=BF
BE
=
BF
. In the triangle
E
B
C
EBC
EBC
, let N be the foot of the altitude relative to
E
C
EC
EC
. Let
G
G
G
be the intersection between
A
D
AD
A
D
and the extension of the previously mentioned altitude.
F
G
FG
FG
and
E
C
EC
EC
intersect at point
P
P
P
, and the lines
N
F
NF
NF
and
D
C
DC
D
C
intersect at point
T
T
T
. Prove that the line
D
P
DP
D
P
is perpendicular to the line
B
T
BT
BT
.
1
1
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Rioplatense Olympiad 2013, Level 3, Problem 1
Let
a
,
b
,
c
,
d
a,b,c,d
a
,
b
,
c
,
d
be real positive numbers such that
a
2
+
b
2
+
c
2
+
d
2
=
1
a^2+b^2+c^2+d^2 = 1
a
2
+
b
2
+
c
2
+
d
2
=
1
. Prove that
(
1
−
a
)
(
1
−
b
)
(
1
−
c
)
(
1
−
d
)
≥
a
b
c
d
(1-a)(1-b)(1-c)(1-d) \geq abcd
(
1
−
a
)
(
1
−
b
)
(
1
−
c
)
(
1
−
d
)
≥
ab
c
d
.
5
1
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Divisibility on 2 numbers with same amount of digits.
Find all positive integers
n
n
n
for which there exist two distinct numbers of
n
n
n
digits,
a
1
a
2
…
a
n
‾
\overline{a_1a_2\ldots a_n}
a
1
a
2
…
a
n
and
b
1
b
2
…
b
n
‾
\overline{b_1b_2\ldots b_n}
b
1
b
2
…
b
n
, such that the number of
2
n
2n
2
n
digits
a
1
a
2
…
a
n
b
1
b
2
…
b
n
‾
\overline{a_1a_2\ldots a_nb_1b_2\ldots b_n}
a
1
a
2
…
a
n
b
1
b
2
…
b
n
is divisible by
b
1
b
2
…
b
n
a
1
a
2
…
a
n
‾
\overline{b_1b_2\ldots b_na_1a_2\ldots a_n}
b
1
b
2
…
b
n
a
1
a
2
…
a
n
.