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Problems
Contests
National and Regional Contests
Mexico Contests
Regional Olympiad of Mexico Southeast
2018 Regional Olympiad of Mexico Southeast
2018 Regional Olympiad of Mexico Southeast
Part of
Regional Olympiad of Mexico Southeast
Subcontests
(6)
6
1
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find polynomials
Find all polynomials
p
(
x
)
p(x)
p
(
x
)
such that for all reals
a
,
b
a, b
a
,
b
and
c
c
c
, with
a
+
b
+
c
=
0
a+b+c=0
a
+
b
+
c
=
0
, satisfies
p
(
a
3
)
+
p
(
b
3
)
+
p
(
c
3
)
=
3
p
(
a
b
c
)
p(a^3)+p(b^3)+p(c^3)=3p(abc)
p
(
a
3
)
+
p
(
b
3
)
+
p
(
c
3
)
=
3
p
(
ab
c
)
5
1
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Prove tangency
Let
A
B
C
ABC
A
BC
an isosceles triangle with
C
A
=
C
B
CA=CB
C
A
=
CB
and
Γ
\Gamma
Γ
it´s circumcircle. The perpendicular to
C
B
CB
CB
through
B
B
B
intersect
Γ
\Gamma
Γ
in points
B
B
B
and
E
E
E
. The parallel to
B
C
BC
BC
through
A
A
A
intersect
Γ
\Gamma
Γ
in points
A
A
A
and
D
D
D
. Let
F
F
F
the intersection of
E
D
ED
E
D
and
B
C
,
I
BC, I
BC
,
I
the intersection of
B
D
BD
B
D
and
E
C
,
Ω
EC, \Omega
EC
,
Ω
the cricumcircle of the triangle
A
D
I
ADI
A
D
I
and
Φ
\Phi
Φ
the circumcircle of
B
E
F
BEF
BEF
.If
O
O
O
and
P
P
P
are the centers of
Γ
\Gamma
Γ
and
Φ
\Phi
Φ
, respectively, prove that
O
P
OP
OP
is tangent to
Ω
\Omega
Ω
4
1
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an infinite sequence
For every natural
n
n
n
let
a
n
=
20
…
018
a_n=20\dots 018
a
n
=
20
…
018
with
n
n
n
ceros, for example,
a
1
=
2018
,
a
3
=
200018
,
a
7
=
2000000018
a_1=2018, a_3=200018, a_7=2000000018
a
1
=
2018
,
a
3
=
200018
,
a
7
=
2000000018
. Prove that there are infinity values of
n
n
n
such that
2018
2018
2018
divides
a
n
a_n
a
n
3
1
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prove a quadrilateral is cyclic
Let
A
B
C
ABC
A
BC
a triangle with circumcircle
Γ
\Gamma
Γ
and
R
R
R
a point inside
A
B
C
ABC
A
BC
such that
∠
A
B
R
=
∠
R
B
C
\angle ABR=\angle RBC
∠
A
BR
=
∠
RBC
. Let
Γ
1
\Gamma_1
Γ
1
and
Γ
2
\Gamma_2
Γ
2
the circumcircles of triangles
A
R
B
ARB
A
RB
and
C
R
B
CRB
CRB
respectly. The parallel to
A
C
AC
A
C
that pass through
R
R
R
, intersect
Γ
\Gamma
Γ
in
D
D
D
and
E
E
E
, with
D
D
D
on the same side of
B
R
BR
BR
that
A
A
A
and
E
E
E
on the same side of
B
R
BR
BR
that
C
C
C
.
A
D
AD
A
D
intersect
Γ
1
\Gamma_1
Γ
1
in
P
P
P
and
C
E
CE
CE
intersect
Γ
2
\Gamma_2
Γ
2
in
Q
Q
Q
. Prove that
A
P
Q
C
APQC
A
PQC
is cyclic if and only if
A
B
=
B
C
AB=BC
A
B
=
BC
2
1
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prove divisibility
Let
n
=
2
2018
−
1
3
n=\frac{2^{2018}-1}{3}
n
=
3
2
2018
−
1
. Prove that
n
n
n
divides
2
n
−
2
2^n-2
2
n
−
2
.
1
1
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Found winning strategy
Lalo and Sergio play in a regular polygon of
n
≥
4
n\geq 4
n
≥
4
sides. In his turn, Lalo paints a diagonal or side of pink, and in his turn Sergio paint a diagonal or side of orange. Wins the game who achieve paint the three sides of a triangle with his color, if none of the players can win, they game tie. Lalo starts playing. Determines all natural numbers
n
n
n
such that one of the players have winning strategy.