MathDB
Problems
Contests
National and Regional Contests
Mexico Contests
Regional Olympiad of Mexico Center Zone
2012 Regional Olympiad of Mexico Center Zone
2012 Regional Olympiad of Mexico Center Zone
Part of
Regional Olympiad of Mexico Center Zone
Subcontests
(6)
3
1
Hide problems
congruent triangles wanted, # with 60^o
In the parallelogram
A
B
C
D
ABCD
A
BC
D
,
∠
B
A
D
=
6
0
∘
\angle BAD =60 ^ \circ
∠
B
A
D
=
6
0
∘
. Let
E
E
E
be the intersection point of the diagonals. The circle circumscribed to the triangle
A
C
D
ACD
A
C
D
intersects the line
A
B
AB
A
B
at the point
K
K
K
(different from
A
A
A
), the line
B
D
BD
B
D
at the point
P
P
P
(different from
D
D
D
), and to the line
B
C
BC
BC
in
L
L
L
(different from
C
C
C
). The line
E
P
EP
EP
intersects the circumscribed circle of the triangle
C
E
L
CEL
CE
L
at the points
E
E
E
and
M
M
M
. Show that the triangles
K
L
M
KLM
K
L
M
and
C
A
P
CAP
C
A
P
are congruent.
5
1
Hide problems
A cool sum
Consider and odd prime
p
p
p
. For each
i
i
i
at
{
1
,
2
,
.
.
.
,
p
−
1
}
\{1, 2,..., p-1\}
{
1
,
2
,
...
,
p
−
1
}
, let
r
i
r_i
r
i
be the rest of
i
p
i^p
i
p
when it is divided by
p
2
p^2
p
2
. Find the sum:
r
1
+
r
2
+
.
.
.
+
r
p
−
1
r_1 + r_2 + ... + r_{p-1}
r
1
+
r
2
+
...
+
r
p
−
1
1
1
Hide problems
Easy combinatorics
Consider the set:
A
=
{
1
,
2
,
.
.
.
,
100
}
A = \{1, 2,..., 100\}
A
=
{
1
,
2
,
...
,
100
}
Prove that if we take
11
11
11
different elements from
A
A
A
, there are
x
,
y
x, y
x
,
y
such that
x
≠
y
x \neq y
x
=
y
and
0
<
∣
x
−
y
∣
<
1
0 < |\sqrt{x} - \sqrt{y}| < 1
0
<
∣
x
−
y
∣
<
1
2
1
Hide problems
Mexico Regional Olympiad 2012.
Let
m
,
n
m, n
m
,
n
integers such that:
(
n
−
1
)
3
+
n
3
+
(
n
+
1
)
3
=
m
3
(n-1)^3+n^3+(n+1)^3=m^3
(
n
−
1
)
3
+
n
3
+
(
n
+
1
)
3
=
m
3
Prove that 4 divides
n
n
n
4
1
Hide problems
Mexico Regional Olympiad. 2012.
On an acute triangle
A
B
C
ABC
A
BC
we draw the internal bisector of
<
A
B
C
<ABC
<
A
BC
,
B
E
BE
BE
, and the altitude
A
D
AD
A
D
, (
D
D
D
on
B
C
BC
BC
), show that
<
C
D
E
<CDE
<
C
D
E
it's bigger than 45 degrees.
6
1
Hide problems
Regional Olympiad in Mexico.
A board of
2
n
2n
2
n
x
2
n
2n
2
n
is colored chess style, a movement is the changing of colors of a
2
2
2
x
2
2
2
square. For what integers
n
n
n
is possible to complete the board with one color using a finite number of movements?