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Problems
Contests
National and Regional Contests
Mexico Contests
Regional Olympiad of Mexico Center Zone
2015 Regional Olympiad of Mexico Center Zone
2015 Regional Olympiad of Mexico Center Zone
Part of
Regional Olympiad of Mexico Center Zone
Subcontests
(6)
6
1
Hide problems
\pi (a + b + c)^2 <=12T
We have
3
3
3
circles such that any
2
2
2
of them are externally tangent. Let
a
a
a
be length of the outer tangent common to a pair of them. The lengths
b
b
b
and
c
c
c
are defined similarly. If
T
T
T
is the sum of the areas of such circles, show that
π
(
a
+
b
+
c
)
2
≤
12
T
\pi (a + b + c)^2 \le 12T
π
(
a
+
b
+
c
)
2
≤
12
T
.Note: In In the case of externally tangent circles, the common external tangent is the segment tangent to them that touches them at different points.
5
1
Hide problems
line connecting centers of triangles DEN, DFM is perpendicular to AD
In the triangle
A
B
C
ABC
A
BC
, we have that
M
M
M
and
N
N
N
are points on
A
B
AB
A
B
and
A
C
AC
A
C
, respectively, such that
B
C
BC
BC
is parallel to
M
N
MN
MN
. A point
D
D
D
is chosen inside the triangle
A
M
N
AMN
A
MN
. Let
E
E
E
and
F
F
F
be the points of intersection of
M
N
MN
MN
with
B
D
BD
B
D
and
C
D
CD
C
D
, respectively. Show that the line joining the centers of the circles circumscribed to the triangles
D
E
N
DEN
D
EN
and
D
F
M
DFM
D
FM
is perpendicular to
A
D
AD
A
D
.
3
1
Hide problems
black squares in 2015 x2015 board
A board of size
2015
×
2015
2015 \times 2015
2015
×
2015
is covered with sub-boards of size
2
×
2
2 \times 2
2
×
2
, each of which is painted like chessboard. Each sub-board covers exactly
4
4
4
squares of the board and each square of the board is covered with at least one square of a sub-board (the painted of the sub-boards can be of any shape). Prove that there is a way to cover the board in such a way that there are exactly
2015
2015
2015
black squares visible. What is the maximum number of visible black squares?
2
1
Hide problems
angle bisector is wanted, circle through A tangent to BC related
In the triangle
A
B
C
ABC
A
BC
, we have that
∠
B
A
C
\angle BAC
∠
B
A
C
is acute. Let
Γ
\Gamma
Γ
be the circle that passes through
A
A
A
and is tangent to the side
B
C
BC
BC
at
C
C
C
. Let
M
M
M
be the midpoint of
B
C
BC
BC
and let
D
D
D
be the other point of intersection of
Γ
\Gamma
Γ
with
A
M
AM
A
M
. If
B
D
BD
B
D
cuts back to
Γ
\Gamma
Γ
at
E
E
E
, show that
A
C
AC
A
C
is the bisector of
∠
B
A
E
\angle BAE
∠
B
A
E
.
1
1
Hide problems
magic square 3x3 from one 9 element block of first 360 naturals
The first
360
360
360
natural numbers are separated into
9
9
9
blocks in such a way that the numbers in each block are consecutive. Then, the numbers in each block are added, obtaining
9
9
9
numbers. Is it possible to fill a
3
×
3
3 \times 3
3
×
3
grid and form a magic square with these numbers? Note: In a magic square, the sum of the numbers written in any column, diagonal or row of the grid is the same.
4
1
Hide problems
prime condition
Find all natural integers
m
,
n
m, n
m
,
n
such that
m
,
2
+
m
,
2
n
+
m
,
2
+
2
n
+
m
m, 2+m, 2^n+m, 2+2^n+m
m
,
2
+
m
,
2
n
+
m
,
2
+
2
n
+
m
are all prime numbers