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Problems
Contests
National and Regional Contests
Mexico Contests
Regional Olympiad of Mexico Center Zone
2018 Regional Olympiad of Mexico Center Zone
2018 Regional Olympiad of Mexico Center Zone
Part of
Regional Olympiad of Mexico Center Zone
Subcontests
(6)
4
1
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2 player game on a n x n board
Ana and Natalia alternately play on a
n
×
n
n \times n
n
×
n
board (Ana rolls first and
n
>
1
n> 1
n
>
1
). At the beginning, Ana's token is placed in the upper left corner and Natalia's in the lower right corner. A turn consists of moving the corresponding piece in any of the four directions (it is not allowed to move diagonally), without leaving the board. The winner is whoever manages to place their token on the opponent's token. Determine if either of them can secure victory after a finite number of turns.
6
1
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exists triangle with sides D' P , E'Q and F'R , altitudes related
Let
△
A
B
C
\vartriangle ABC
△
A
BC
be a triangle with orthocenter
H
H
H
and altitudes
A
D
AD
A
D
,
B
E
BE
BE
and
C
F
CF
CF
. Let
D
′
D'
D
′
,
E
′
E'
E
′
and
F
′
F'
F
′
be the intersections of the heights
A
D
AD
A
D
,
B
E
BE
BE
and
C
F
CF
CF
, respectively, with the circumcircle of
△
A
B
C
\vartriangle ABC
△
A
BC
, so that they are different points from the vertices of triangle
△
A
B
C
\vartriangle ABC
△
A
BC
. Let
L
L
L
,
M
M
M
and
N
N
N
be the midpoints of
B
C
BC
BC
,
A
C
AC
A
C
and
A
B
AB
A
B
, respectively. Let
P
P
P
,
Q
Q
Q
and
R
R
R
be the intersections of the circumcircle with
L
H
LH
L
H
,
M
H
MH
M
H
and
N
H
NH
N
H
, respectively, such that
P
P
P
and
A
A
A
are on opposite sides of
B
C
BC
BC
,
Q
Q
Q
and
A
A
A
are on opposite sides of
A
C
AC
A
C
and
R
R
R
and
C
C
C
are on opposite sides of
A
B
AB
A
B
. Show that there exists a triangle whose sides have the lengths of the segments
D
′
P
D' P
D
′
P
,
E
′
Q
E'Q
E
′
Q
, and
F
′
R
F'R
F
′
R
.
2
1
Hide problems
AP = PC wanted, circumcircle and 2 midpoints
Let
△
A
B
C
\vartriangle ABC
△
A
BC
be a triangle and let
Γ
\Gamma
Γ
its circumscribed circle. Let
M
M
M
be the midpoint of the side
B
C
BC
BC
and let
D
D
D
be the point of intersection of the line
A
M
AM
A
M
with
Γ
\Gamma
Γ
. By
D
D
D
a straight line is drawn parallel to
B
C
BC
BC
, which intersects
Γ
\Gamma
Γ
at a point
E
E
E
. Let
N
N
N
be the midpoint of the segment
A
E
AE
A
E
and let
P
P
P
be the point of intersection of
C
N
CN
CN
with
A
M
AM
A
M
. Show that
A
P
=
P
C
AP = PC
A
P
=
PC
.
5
1
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p^2 + q^2 + 49r^2 = 9k^2-101
Find all solutions of the equation
p
2
+
q
2
+
49
r
2
=
9
k
2
−
101
p ^ 2 + q ^ 2 + 49r ^ 2 = 9k ^ 2-101
p
2
+
q
2
+
49
r
2
=
9
k
2
−
101
with
p
p
p
,
q
q
q
and
r
r
r
positive prime numbers and
k
k
k
a positive integer.
3
1
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numbering k from n lines in general position
Consider
n
n
n
lines in the plane in general position, that is, there are not three of the
n
n
n
lines that pass through the same point. Determine if it is possible to label the
k
k
k
points where these lines are inserted with the numbers
1
1
1
through
k
k
k
(using each number exactly once), so that on each line, the labels of the
n
−
1
n-1
n
−
1
points of that line are arranged in increasing order (in one of the two directions in which they can be traversed).
1
1
Hide problems
N-M where M,N two 5-digit ''consecutive'' palindromes
Let
M
M
M
and
N
N
N
be two positive five-digit palindrome integers, such that
M
<
N
M <N
M
<
N
and there is no other palindrome number between them. Determine the possible values of
N
−
M
N-M
N
−
M
.