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National and Regional Contests
Ecuador Contests
Ecuador Mathematical Olympiad (OMEC)
2016 Ecuador NMO (OMEC)
2016 Ecuador NMO (OMEC)
Part of
Ecuador Mathematical Olympiad (OMEC)
Subcontests
(3)
6
1
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sum c_ix_i is not divisible by n^3 2016 Ecuador NMO (OMEC) 3.6
A positive integer
n
n
n
is "olympic" if there are
n
n
n
non-negative integers
x
1
,
x
2
,
.
.
.
,
x
n
x_1, x_2, ..., x_n
x
1
,
x
2
,
...
,
x
n
that satisfy that:
∙
\bullet
∙
There is at least one positive integer
j
j
j
:
1
≤
j
≤
n
1 \le j \le n
1
≤
j
≤
n
such that
x
j
≠
0
x_j \ne 0
x
j
=
0
.
∙
\bullet
∙
For any way of choosing
n
n
n
numbers
c
1
,
c
2
,
.
.
.
,
c
n
c_1, c_2, ..., c_n
c
1
,
c
2
,
...
,
c
n
from the set
{
−
1
,
0
,
1
}
\{-1, 0, 1\}
{
−
1
,
0
,
1
}
, where not all
c
i
c_i
c
i
are equal to zero, it is true that the sum
c
1
x
1
+
c
2
x
2
+
.
.
.
+
c
n
x
n
c_1x_1 + c_2x_2 +... + c_nx_n
c
1
x
1
+
c
2
x
2
+
...
+
c
n
x
n
is not divisible by
n
3
n^3
n
3
. Find the largest positive "olympic" integer.
3
1
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<MBN=? AB=BC=CD, equilaterals MAP, NQD 2016 Ecuador NMO (OMEC) 3.3
Let
A
,
B
,
C
,
D
A, B, C, D
A
,
B
,
C
,
D
be four different points on a line
ℓ
\ell
ℓ
, such that
A
B
=
B
C
=
C
D
AB = BC = CD
A
B
=
BC
=
C
D
. In one of the semiplanes determined by the line
ℓ
\ell
ℓ
, the points
P
P
P
and
Q
Q
Q
are chosen in such a way that the triangle
C
P
Q
CPQ
CPQ
is equilateral with its vertices named clockwise. Let
M
M
M
and
N
N
N
be two points on the plane such that the triangles
M
A
P
MAP
M
A
P
and
N
Q
D
NQD
NQ
D
are equilateral (the vertices are also named clockwise). Find the measure of the angle
∠
M
B
N
\angle MBN
∠
MBN
.
2
1
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even no of diagonals in convex 2017-gon 2016 Ecuador NMO (OMEC) 3.2
All diagonals are plotted in a
2017
2017
2017
-sided convex polygon. A line
ℓ
\ell
ℓ
intersects said polygon but does not pass through any of its vertices. Show that the line
ℓ
\ell
ℓ
intersects an even number of diagonals of said polygon.