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Contests
National and Regional Contests
Ecuador Contests
Ecuador Mathematical Olympiad (OMEC)
2021 Ecuador NMO (OMEC)
2021 Ecuador NMO (OMEC)
Part of
Ecuador Mathematical Olympiad (OMEC)
Subcontests
(6)
6
1
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Diophantine equation with HIGH values
Find all positive integers
a
,
b
,
c
a, b, c
a
,
b
,
c
such that
a
b
+
1
ab+1
ab
+
1
and
c
c
c
are coprimes and:
a
(
b
a
+
1
)
(
c
a
2
+
b
a
+
1
)
=
202
1
2021
a(ba+1)(ca^2+ba+1)=2021^{2021}
a
(
ba
+
1
)
(
c
a
2
+
ba
+
1
)
=
202
1
2021
5
1
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Integer triangle construction
Find an acutangle triangle such that its sides and altitudes have integer length.
4
1
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If it has chess, it has combinatorics
In a board
8
8
8
x
8
8
8
, the unit squares have numbers
1
−
64
1-64
1
−
64
, as shown. The unit square with a multiple of
3
3
3
on it are red. Find the minimum number of chess' bishops that we need to put on the board such that any red unit square either has a bishop on it or is attacked by at least one bishop. Note: A bishops moves diagonally. https://i.imgur.com/03baBwp.jpeg
3
1
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Geometry, Locus and moving circumferences
Let
T
1
T_1
T
1
and
T
2
T_2
T
2
internally tangent circumferences at
P
P
P
, with radius
R
R
R
and
2
R
2R
2
R
, respectively. Find the locus traced by
P
P
P
as
T
1
T_1
T
1
rolls tangentially along the entire perimeter of
T
2
T_2
T
2
2
1
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Polynomial square and cube at the same time
Let
P
(
x
)
P(x)
P
(
x
)
a grade 3 polynomial such that:
P
(
1
)
=
1
,
P
(
2
)
=
4
,
P
(
3
)
=
9
P(1)=1, P(2)=4, P(3)=9
P
(
1
)
=
1
,
P
(
2
)
=
4
,
P
(
3
)
=
9
Find the value of
P
(
10
)
+
P
(
−
6
)
P(10)+P(-6)
P
(
10
)
+
P
(
−
6
)
1
1
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Classic attack for this problem
Find all integers
n
n
n
such that
4
n
n
2
+
3
\frac{4n}{n^2 +3 }
n
2
+
3
4
n
is an integer.