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Problems
Contests
National and Regional Contests
Mexico Contests
Regional Olympiad of Mexico Southeast
2017 Regional Olympiad of Mexico Southeast
2017 Regional Olympiad of Mexico Southeast
Part of
Regional Olympiad of Mexico Southeast
Subcontests
(6)
6
1
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Fibonacci sequence
Consider
f
1
=
1
,
f
2
=
1
f_1=1, f_2=1
f
1
=
1
,
f
2
=
1
and
f
n
+
1
=
f
n
+
f
n
−
1
f_{n+1}=f_n+f_{n-1}
f
n
+
1
=
f
n
+
f
n
−
1
for
n
≥
2
n\geq 2
n
≥
2
. Determine if exists
n
≤
1000001
n\leq 1000001
n
≤
1000001
such that the last three digits of
f
n
f_n
f
n
are zero.
5
1
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circumcenter and orthocenter
Consider an acutangle triangle
A
B
C
ABC
A
BC
with circumcenter
O
O
O
. A circumference that passes through
B
B
B
and
O
O
O
intersect sides
B
C
BC
BC
and
A
B
AB
A
B
in points
P
P
P
and
Q
Q
Q
. Prove that the orthocenter of triangle
O
P
Q
OPQ
OPQ
is on
A
C
AC
A
C
.
4
1
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diophantin equation with factorial
Find all couples of positive integers
m
m
m
and
n
n
n
such that
n
!
+
5
=
m
3
n!+5=m^3
n
!
+
5
=
m
3
3
1
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divisible in primes
Let
p
p
p
of prime of the form
3
k
+
2
3k+2
3
k
+
2
such that
a
2
+
a
b
+
b
2
a^2+ab+b^2
a
2
+
ab
+
b
2
is divisible by
p
p
p
for some integers
a
a
a
and
b
b
b
. Prove that both of
a
a
a
and
b
b
b
are divisible by
p
p
p
.
2
1
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games in a tournament
In the Cancun´s league participate
30
30
30
teams. For this tournament we want to divide the
30
30
30
teams in
2
2
2
groups such that:
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
1.
<
/
s
p
a
n
>
<span class='latex-bold'>1.</span>
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
1.
<
/
s
p
an
>
Every team plays exactly
82
82
82
games
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
2.
<
/
s
p
a
n
>
<span class='latex-bold'>2.</span>
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
2.
<
/
s
p
an
>
The number of gamen between teams of differents groups is equal to the half of games played.Can we do this?
1
1
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Prove a point is fixed
Let
A
B
C
ABC
A
BC
a triangle and
C
C
C
it´s circuncircle. Let
D
D
D
a point in arc
A
B
AB
A
B
that not contain
A
A
A
, diferent of
B
B
B
and
C
C
C
such that
C
D
CD
C
D
and
A
B
AB
A
B
are not parallel. Let
E
E
E
the intersection of
C
D
CD
C
D
and
A
B
AB
A
B
and
O
O
O
the circumcircle of triangle
D
B
E
DBE
D
BE
. Prove that the measure of
∠
O
B
E
\angle OBE
∠
OBE
does not depend of the choice of
D
D
D
.