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Problems
Contests
International Contests
Cono Sur Olympiad
2015 Cono Sur Olympiad
2015 Cono Sur Olympiad
Part of
Cono Sur Olympiad
Subcontests
(6)
6
1
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Subsets said to be friends (2015 OMCS #6)
Let
S
=
{
1
,
2
,
3
,
…
,
2046
,
2047
,
2048
}
S = \{1, 2, 3, \ldots , 2046, 2047, 2048\}
S
=
{
1
,
2
,
3
,
…
,
2046
,
2047
,
2048
}
. Two subsets
A
A
A
and
B
B
B
of
S
S
S
are said to be friends if the following conditions are true: [*] They do not share any elements. [*] They both have the same number of elements. [*] The product of all elements from
A
A
A
equals the product of all elements from
B
B
B
.Prove that there are two subsets of
S
S
S
that are friends such that each one of them contains at least
738
738
738
elements.
5
1
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Infinite sequence of positive integers (2015 OMCS #5)
Determine if there exists an infinite sequence of not necessarily distinct positive integers
a
1
,
a
2
,
a
3
,
…
a_1, a_2, a_3,\ldots
a
1
,
a
2
,
a
3
,
…
such that for any positive integers
m
m
m
and
n
n
n
where
1
≤
m
<
n
1 \leq m < n
1
≤
m
<
n
, the number
a
m
+
1
+
a
m
+
2
+
…
+
a
n
a_{m+1} + a_{m+2} + \ldots + a_{n}
a
m
+
1
+
a
m
+
2
+
…
+
a
n
is not divisible by
a
1
+
a
2
+
…
+
a
m
a_1 + a_2 + \ldots + a_m
a
1
+
a
2
+
…
+
a
m
.
4
1
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Show that a quadrilateral is a square (2015 OMCS #4)
Let
A
B
C
D
ABCD
A
BC
D
be a convex quadrilateral such that
∠
B
A
D
=
9
0
∘
\angle{BAD} = 90^{\circ}
∠
B
A
D
=
9
0
∘
and its diagonals
A
C
AC
A
C
and
B
D
BD
B
D
are perpendicular. Let
M
M
M
be the midpoint of side
C
D
CD
C
D
, and
E
E
E
be the intersection of
B
M
BM
BM
and
A
C
AC
A
C
. Let
F
F
F
be a point on side
A
D
AD
A
D
such that
B
M
BM
BM
and
E
F
EF
EF
are perpendicular. If
C
E
=
A
F
2
CE = AF\sqrt{2}
CE
=
A
F
2
and
F
D
=
C
E
2
FD = CE\sqrt{2}
F
D
=
CE
2
, show that
A
B
C
D
ABCD
A
BC
D
is a square.
2
1
Hide problems
3n coloured lines on the plane (2015 OMCS #2)
3
n
3n
3
n
lines are drawn on the plane (
n
>
1
n > 1
n
>
1
), such that no two of them are parallel and no three of them are concurrent. Prove that, if
2
n
2n
2
n
of the lines are coloured red and the other
n
n
n
lines blue, there are at least two regions of the plane such that all of their borders are red.Note: for each region, all of its borders are contained in the original set of lines, and no line passes through the region.
1
1
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Divisibility by 81 (2015 OMCS #1)
Show that, for any integer
n
n
n
, the number
n
3
−
9
n
+
27
n^3 - 9n + 27
n
3
−
9
n
+
27
is not divisible by
81
81
81
.
3
1
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Find the lenght
Given a acute triangle
P
A
1
B
1
PA_1B_1
P
A
1
B
1
is inscribed in the circle
Γ
\Gamma
Γ
with radius
1
1
1
. for all integers
n
≥
1
n \ge 1
n
≥
1
are defined:
C
n
C_n
C
n
the foot of the perpendicular from
P
P
P
to
A
n
B
n
A_nB_n
A
n
B
n
O
n
O_n
O
n
is the center of
⊙
(
P
A
n
B
n
)
\odot (PA_nB_n)
⊙
(
P
A
n
B
n
)
A
n
+
1
A_{n+1}
A
n
+
1
is the foot of the perpendicular from
C
n
C_n
C
n
to
P
A
n
PA_n
P
A
n
B
n
+
1
≡
P
B
n
∩
O
n
A
n
+
1
B_{n+1} \equiv PB_n \cap O_nA_{n+1}
B
n
+
1
≡
P
B
n
∩
O
n
A
n
+
1
If
P
C
1
=
2
PC_1 =\sqrt{2}
P
C
1
=
2
, find the length of
P
O
2015
PO_{2015}
P
O
2015
Cono Sur Olympiad - 2015 - Day 1 - Problem 3