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Problems
Contests
International Contests
Cono Sur Olympiad
2018 Cono Sur Olympiad
2018 Cono Sur Olympiad
Part of
Cono Sur Olympiad
Subcontests
(6)
6
1
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Sixth problem
A sequence
a
1
,
a
2
,
…
,
a
n
a_1, a_2,\dots, a_n
a
1
,
a
2
,
…
,
a
n
of positive integers is alagoana, if for every
n
n
n
positive integer, one have these two conditions I-
a
n
!
=
a
1
⋅
a
2
⋅
a
3
⋯
a
n
a_{n!} = a_1\cdot a_2\cdot a_3\cdots a_n
a
n
!
=
a
1
⋅
a
2
⋅
a
3
⋯
a
n
II- The number
a
n
a_n
a
n
is the
n
n
n
-power of a positive integer. Find all the sequence(s) alagoana.
5
1
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Fifth problem
Let
A
B
C
ABC
A
BC
be an acute-angled triangle with
∠
B
A
C
=
6
0
∘
\angle BAC = 60^{\circ}
∠
B
A
C
=
6
0
∘
and with incenter
I
I
I
and circumcenter
O
O
O
. Let
H
H
H
be the point diametrically opposite(antipode) to
O
O
O
in the circumcircle of
△
B
O
C
\triangle BOC
△
BOC
. Prove that
I
H
=
B
I
+
I
C
IH=BI+IC
I
H
=
B
I
+
I
C
.
4
1
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Fourth problem
For each interger
n
≥
4
n\geq 4
n
≥
4
, we consider the
m
m
m
subsets
A
1
,
A
2
,
…
,
A
m
A_1, A_2,\dots, A_m
A
1
,
A
2
,
…
,
A
m
of
{
1
,
2
,
3
,
…
,
n
}
\{1, 2, 3,\dots, n\}
{
1
,
2
,
3
,
…
,
n
}
, such that
A
1
A_1
A
1
has exactly one element,
A
2
A_2
A
2
has exactly two elements,....,
A
m
A_m
A
m
has exactly
m
m
m
elements and none of these subsets is contained in any other set. Find the maximum value of
m
m
m
.
3
1
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Third problem
Define the product
P
n
=
1
!
⋅
2
!
⋅
3
!
⋯
(
n
−
1
)
!
⋅
n
!
P_n=1! \cdot 2!\cdot 3!\cdots (n-1)!\cdot n!
P
n
=
1
!
⋅
2
!
⋅
3
!
⋯
(
n
−
1
)!
⋅
n
!
a) Find all positive integers
m
m
m
, such that
P
2020
m
!
\frac {P_{2020}}{m!}
m
!
P
2020
is a perfect square. b) Prove that there are infinite many value(s) of
n
n
n
, such that
P
n
m
!
\frac {P_{n}}{m!}
m
!
P
n
is a perfect square, for at least two positive integers
m
m
m
.
2
1
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Second problem
Prove that every positive integer can be formed by the sums of powers of 3, 4 and 7, where do not appear two powers of the same number and with the same exponent. Example:
2
=
7
0
+
7
0
2= 7^0 + 7^0
2
=
7
0
+
7
0
and
22
=
3
2
+
3
2
+
4
1
22=3^2 + 3^2+4^1
22
=
3
2
+
3
2
+
4
1
are not valid representations, but
2
=
3
0
+
7
0
2=3^0+7^0
2
=
3
0
+
7
0
and
22
=
3
2
+
3
0
+
4
1
+
4
0
+
7
1
22=3^2+3^0+4^1+4^0+7^1
22
=
3
2
+
3
0
+
4
1
+
4
0
+
7
1
are valid representations.
1
1
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First problem
Let
A
B
C
D
ABCD
A
BC
D
be a convex quadrilateral, where
R
R
R
and
S
S
S
are points in
D
C
DC
D
C
and
A
B
AB
A
B
, respectively, such that
A
D
=
R
C
AD=RC
A
D
=
RC
and
B
C
=
S
A
BC=SA
BC
=
S
A
. Let
P
P
P
,
Q
Q
Q
and
M
M
M
be the midpoints of
R
D
RD
R
D
,
B
S
BS
BS
and
C
A
CA
C
A
, respectively. If
∠
M
P
C
+
∠
M
Q
A
=
90
\angle MPC + \angle MQA = 90
∠
MPC
+
∠
MQ
A
=
90
, prove that
A
B
C
D
ABCD
A
BC
D
is cyclic.