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Problems
Contests
International Contests
Cono Sur Olympiad
2021 Cono Sur Olympiad
2021 Cono Sur Olympiad
Part of
Cono Sur Olympiad
Subcontests
(6)
1
1
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P1 Cono Sur 2021 - Guarani Numbers
We say that a positive integer is guarani if the sum of the number with its reverse is a number that only has odd digits. For example,
249
249
249
and
30
30
30
are guarani, since
249
+
942
=
1191
249 + 942 = 1191
249
+
942
=
1191
and
30
+
03
=
33
30 + 03 = 33
30
+
03
=
33
. a) How many
2021
2021
2021
-digit numbers are guarani? b) How many
2023
2023
2023
-digit numbers are guarani?
6
1
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P6 Cono Sur 2021
Let
A
B
C
ABC
A
BC
be a scalene triangle with circle
Γ
\Gamma
Γ
. Let
P
,
Q
,
R
,
S
P,Q,R,S
P
,
Q
,
R
,
S
distinct points on the
B
C
BC
BC
side, in that order, such that
∠
B
A
P
=
∠
C
A
S
\angle BAP = \angle CAS
∠
B
A
P
=
∠
C
A
S
and
∠
B
A
Q
=
∠
C
A
R
\angle BAQ = \angle CAR
∠
B
A
Q
=
∠
C
A
R
. Let
U
,
V
,
W
,
Z
U, V, W, Z
U
,
V
,
W
,
Z
be the intersections, distinct from
A
A
A
, of the
A
P
,
A
Q
,
A
R
AP, AQ, AR
A
P
,
A
Q
,
A
R
and
A
S
AS
A
S
with
Γ
\Gamma
Γ
, respectively. Let
X
=
U
Q
∩
S
W
X = UQ \cap SW
X
=
U
Q
∩
S
W
,
Y
=
P
V
∩
Z
R
Y = PV \cap ZR
Y
=
P
V
∩
ZR
,
T
=
U
R
∩
V
S
T = UR \cap VS
T
=
U
R
∩
V
S
and
K
=
P
W
∩
Z
Q
K = PW \cap ZQ
K
=
P
W
∩
ZQ
. Suppose that the points
M
M
M
and
N
N
N
are well determined, such that
M
=
K
X
∩
T
Y
M = KX \cap TY
M
=
K
X
∩
T
Y
and
N
=
T
X
∩
K
Y
N = TX \cap KY
N
=
TX
∩
K
Y
. Show that
M
,
N
,
A
M, N, A
M
,
N
,
A
are collinear.
5
1
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P5 Cono Sur 2021
Given an integer
n
≥
3
n \geq 3
n
≥
3
, determine if there are
n
n
n
integers
b
1
,
b
2
,
…
,
b
n
b_1, b_2, \dots , b_n
b
1
,
b
2
,
…
,
b
n
, distinct two-by-two (that is,
b
i
≠
b
j
b_i \neq b_j
b
i
=
b
j
for all
i
≠
j
i \neq j
i
=
j
) and a polynomial
P
(
x
)
P(x)
P
(
x
)
with coefficients integers, such that
P
(
b
1
)
=
b
2
,
P
(
b
2
)
=
b
3
,
…
,
P
(
b
n
−
1
)
=
b
n
P(b_1) = b_2, P(b_2) = b_3, \dots , P(b_{n-1}) = b_n
P
(
b
1
)
=
b
2
,
P
(
b
2
)
=
b
3
,
…
,
P
(
b
n
−
1
)
=
b
n
and
P
(
b
n
)
=
b
1
P(b_n) = b_1
P
(
b
n
)
=
b
1
.
4
1
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P4 Cono Sur 2021
In a heap there are
2021
2021
2021
stones. Two players
A
A
A
and
B
B
B
play removing stones of the pile, alternately starting with
A
A
A
. A valid move for
A
A
A
consists of remove
1
,
2
1, 2
1
,
2
or
7
7
7
stones. A valid move for B is to remove
1
,
3
,
4
1, 3, 4
1
,
3
,
4
or
6
6
6
stones. The player who leaves the pile empty after making a valid move wins. Determine if some of the players have a winning strategy. If such a strategy exists, explain it.
3
1
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P3 Cono Sur 2021
In a tennis club, each member has exactly
k
>
0
k > 0
k
>
0
friends, and a tournament is organized in rounds such that each pair of friends faces each other in matches exactly once. Rounds are played in simultaneous matches, choosing pairs until they cannot choose any more (that is, among the unchosen people, there is not a pair of friends which has its match pending). Determine the maximum number of rounds the tournament can have, depending on
k
k
k
.
2
1
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P2 Cono Sur 2021
Let
A
B
C
ABC
A
BC
be a triangle and
I
I
I
its incenter. The lines
B
I
BI
B
I
and
C
I
CI
C
I
intersect the circumcircle of
A
B
C
ABC
A
BC
again at
M
M
M
and
N
N
N
, respectively. Let
C
1
C_1
C
1
and
C
2
C_2
C
2
be the circumferences of diameters
N
I
NI
N
I
and
M
I
MI
M
I
, respectively. The circle
C
1
C_1
C
1
intersects
A
B
AB
A
B
at
P
P
P
and
Q
Q
Q
, and the circle
C
2
C_2
C
2
intersects
A
C
AC
A
C
at
R
R
R
and
S
S
S
. Show that
P
P
P
,
Q
Q
Q
,
R
R
R
and
S
S
S
are concyclic.