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Contests
National and Regional Contests
Ecuador Contests
Ecuador Mathematical Olympiad (OMEC)
2020 Ecuador NMO (OMEC)
2020 Ecuador NMO (OMEC)
Part of
Ecuador Mathematical Olympiad (OMEC)
Subcontests
(6)
6
1
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Guayaco board combi nt problem
A board
1
1
1
x
k
k
k
is called guayaco if: -Each unit square is painted with exactly one of
k
k
k
available colors. -If
g
c
d
(
i
,
k
)
>
1
gcd(i,k)>1
g
c
d
(
i
,
k
)
>
1
, the
i
i
i
th unit square is painted with the same color as
(
i
−
1
)
(i-1)
(
i
−
1
)
th unit square. -If
g
c
d
(
i
,
k
)
=
1
gcd(i, k)=1
g
c
d
(
i
,
k
)
=
1
, the
i
i
i
th unit square is painted with the same color as
(
k
−
i
)
(k-i)
(
k
−
i
)
th unit square. Sebastian chooses a positive integer
a
a
a
and calculates the number of boards
1
1
1
x
a
a
a
that are guayacos. After that, David chooses a positive integer
b
b
b
and calculates the number of boards
1
1
1
x
b
b
b
that are guayacos. David wins if the number of boards
1
1
1
x
a
a
a
that are guayacos is the same as the number of boards
1
1
1
x
b
b
b
that are guayacos, otherwise, Sebastian wins. Find all the pairs
(
a
,
b
)
(a,b)
(
a
,
b
)
such that, with those numbers, David wins.
5
1
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Proving inequality geometry problem
In triangle
A
B
C
ABC
A
BC
,
D
D
D
is the middle point of side
B
C
BC
BC
and
M
M
M
is a point on segment
A
D
AD
A
D
such that
A
M
=
3
M
D
AM=3MD
A
M
=
3
M
D
. The barycenter of
A
B
C
ABC
A
BC
and
M
M
M
are on the inscribed circumference of
A
B
C
ABC
A
BC
. Prove that
A
B
+
A
C
>
3
B
C
AB+AC>3BC
A
B
+
A
C
>
3
BC
.
4
1
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Polynomial equation with specific numbers
Find all polynomials
P
(
x
)
P(x)
P
(
x
)
such that, for all real numbers
x
,
y
,
z
x, y, z
x
,
y
,
z
that satisfy
x
+
y
+
z
=
0
x+ y +z =0
x
+
y
+
z
=
0
,
P
(
x
)
+
P
(
y
)
+
P
(
z
)
=
0
P(x) +P(y) +P(z)=0
P
(
x
)
+
P
(
y
)
+
P
(
z
)
=
0
3
1
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Finding two circumcenters
Let
A
B
C
ABC
A
BC
a triangle with circumcircle
Γ
\Gamma
Γ
and circumcenter
O
O
O
. A point
X
X
X
, different from
A
A
A
,
B
B
B
,
C
C
C
, or their diametrically opposite points, on
Γ
\Gamma
Γ
, is chosen. Let
ω
\omega
ω
the circumcircle of
C
O
X
COX
COX
. Let
E
E
E
the second intersection of
X
A
XA
X
A
with
ω
\omega
ω
,
F
F
F
the second intersection of
X
B
XB
XB
with
ω
\omega
ω
and
D
D
D
a point on line
A
B
AB
A
B
such that
C
D
⊥
E
F
CD \perp EF
C
D
⊥
EF
. Prove that
E
E
E
is the circumcenter of
A
D
C
ADC
A
D
C
and
F
F
F
is the circumcenter of
B
D
C
BDC
B
D
C
.
2
1
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Equation with rational integers and no integers numbers
Find all pairs
(
n
,
q
)
(n, q)
(
n
,
q
)
such that
n
n
n
is a positive integer,
q
q
q
is a not integer rational and
n
q
−
q
n^q-q
n
q
−
q
is an integer.
1
1
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Elections Combinatorics Problem
The country OMEC is divided in
5
5
5
regions, each region is divided in
5
5
5
districts, and, in each district,
1001
1001
1001
people vote. Each person choose between
A
A
A
or
B
B
B
. In a district, a candidate's letter wins if it's the letter with the most votes. In a region, a candidate's letter wins if it won in most districts. A candidate is the new president of OMEC if the candidate won in most regions. The candidate
A
A
A
can rearrange the people of each district in each region (for example, A moves someone in District M to District N in region 1), but he can't change them to a different region. Find the minimum number of votes that the candidate
A
A
A
needs to become the new president.