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Problems
Contests
National and Regional Contests
Argentina Contests
Argentina National Olympiad
2020 Argentina National Olympiad
2020 Argentina National Olympiad
Part of
Argentina National Olympiad
Subcontests
(6)
6
1
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2player game, a vertex marked as a trap in a regular n-gon
Let
n
≥
3
n\ge 3
n
≥
3
be an integer. Lucas and Matías play a game in a regular
n
n
n
-sided polygon with a vertex marked as a trap. Initially Matías places a token at one vertex of the polygon. In each step, Lucas says a positive integer and Matías moves the token that number of vertices clockwise or counterclockwise, at his choice.a) Determine all the
n
≥
3
n\ge 3
n
≥
3
such that Matías can locate the token and move it in such a way as to never fall into the trap, regardless of the numbers Lucas says. Give the strategy to Matías.b) Determine all the
n
≥
3
n\ge 3
n
≥
3
such that Lucas can force Matías to fall into the trap. Give the strategy to Lucas.Note. The two players know the value of
n
n
n
and see the polygon.
5
1
Hide problems
max S=a_1a_2a_3 + a_4a_5a_6 +⋯+ a_{2017}a_{2018}a_{2019} + a_{2020}
Determine the highest possible value of:
S
=
a
1
a
2
a
3
+
a
4
a
5
a
6
+
.
.
.
+
a
2017
a
2018
a
2019
+
a
2020
S = a_1a_2a_3 + a_4a_5a_6 +... + a_{2017}a_{2018}a_{2019} + a_{2020}
S
=
a
1
a
2
a
3
+
a
4
a
5
a
6
+
...
+
a
2017
a
2018
a
2019
+
a
2020
where
(
a
1
,
a
2
,
a
3
,
.
.
.
,
a
2020
)
(a_1, a_2, a_3,..., a_{2020})
(
a
1
,
a
2
,
a
3
,
...
,
a
2020
)
is a permutation of
(
1
,
2
,
3
,
.
.
.
,
2020
)
(1,2,3,..., 2020)
(
1
,
2
,
3
,
...
,
2020
)
.Clarification: In
S
S
S
, each term, except the last one, is the multiplication of three numbers.
4
1
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a not prime if (5a^4 + a^2)/(b^4 + 3b^2 + 4) is an integer
Let
a
a
a
and
b
b
b
be positive integers such that
5
a
4
+
a
2
b
4
+
3
b
2
+
4
\frac{5a^4 + a^2}{b^4 + 3b^2 + 4}
b
4
+
3
b
2
+
4
5
a
4
+
a
2
is an integer. Show that
a
a
a
is not prime.
3
1
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angle chasing , <ECB = 30^o, <FBC = 15^o, right isosceles
Let
A
B
C
ABC
A
BC
be a right isosceles triangle with right angle at
A
A
A
. Let
E
E
E
and
F
F
F
be points on A
B
B
B
and
A
C
AC
A
C
respectively such that
∠
E
C
B
=
3
0
o
\angle ECB = 30^o
∠
ECB
=
3
0
o
and
∠
F
B
C
=
1
5
o
\angle FBC = 15^o
∠
FBC
=
1
5
o
. Lines
C
E
CE
CE
and
B
F
BF
BF
intersect at
P
P
P
and line
A
P
AP
A
P
intersects side
B
C
BC
BC
at
D
D
D
. Calculate the measure of angle
∠
F
D
C
\angle FDC
∠
F
D
C
.
2
1
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exactly k black cells n each row and in each column on an nxn board
Let
k
≥
1
k\ge 1
k
≥
1
be an integer. Determine the smallest positive integer
n
n
n
such that some cells on an
n
×
n
n \times n
n
×
n
board can be painted black so that in each row and in each column there are exactly
k
k
k
black cells, and furthermore, the black cells do not share a side or a vertex with another black square.Clarification: You have to answer n based on
k
k
k
.
1
1
Hide problems
171-digit no, 7 divides both S (n) and S (n + 1), sum of digits
For every positive integer
n
n
n
, let
S
(
n
)
S (n)
S
(
n
)
be the sum of the digits of
n
n
n
. Find, if any, a
171
171
171
-digit positive integer
n
n
n
such that
7
7
7
divides
S
(
n
)
S (n)
S
(
n
)
and
7
7
7
divides
S
(
n
+
1
)
S (n + 1)
S
(
n
+
1
)
.