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Problems
Contests
National and Regional Contests
Mexico Contests
Regional Olympiad of Mexico Center Zone
2014 Regional Olympiad of Mexico Center Zone
2014 Regional Olympiad of Mexico Center Zone
Part of
Regional Olympiad of Mexico Center Zone
Subcontests
(5)
4
1
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Rc'/C's = 3/2 wanted, square and a reflection of a point wrt line
Let
A
B
C
D
ABCD
A
BC
D
be a square and let
M
M
M
be the midpoint of
B
C
BC
BC
. Let
C
′
C ^ \prime
C
′
be the reflection of
C
C
C
wrt to
D
M
DM
D
M
. The parallel to
A
B
AB
A
B
passing through
C
′
C ^ \prime
C
′
intersects
A
D
AD
A
D
at
R
R
R
and
B
C
BC
BC
at
S
S
S
. Show that
R
C
′
C
′
S
=
3
2
\frac {RC ^ \prime} {C ^\prime S} = \frac {3} {2}
C
′
S
R
C
′
=
2
3
3
1
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d(O,BC) = inradius of ABC, excircle related
Let
A
B
AB
A
B
be a triangle and
Γ
\Gamma
Γ
the excircle, relative to the vertex
A
A
A
, with center
D
D
D
. The circle
Γ
\Gamma
Γ
is tangent to the lines
A
B
AB
A
B
and
A
C
AC
A
C
at
E
E
E
and
F
F
F
, respectively. Let
P
P
P
and
Q
Q
Q
be the intersections of
E
F
EF
EF
with
B
D
BD
B
D
and
C
D
CD
C
D
, respectively. If
O
O
O
is the point of intersection of
B
Q
BQ
BQ
and
C
P
CP
CP
, show that the distance from
O
O
O
to the line
B
C
BC
BC
is equal to the radius of the inscribed circle in the triangle
A
B
C
ABC
A
BC
.
2
1
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x_1y_1 + x_2y_2 + x_3y_3 <1 if x_1 + y_2 = x_2 + y_3 = x_3 + y_1 =1
Let
x
1
x_1
x
1
,
x
2
x_2
x
2
,
x
3
x_3
x
3
,
y
1
y_1
y
1
,
y
2
y_2
y
2
, and
y
3
y_3
y
3
be positive real numbers, such that
x
1
+
y
2
=
x
2
+
y
3
=
x
3
+
y
1
=
1
x_1 + y_2 = x_2 + y_3 = x_3 + y_1 =1
x
1
+
y
2
=
x
2
+
y
3
=
x
3
+
y
1
=
1
. Prove that
x
1
y
1
+
x
2
y
2
+
x
3
y
3
<
1
x_1y_1 + x_2y_2 + x_3y_3 <1
x
1
y
1
+
x
2
y
2
+
x
3
y
3
<
1
1
1
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product of all positive differences of n different integers divisible by 2014
Find the smallest positive integer
n
n
n
that satisfies that for any
n
n
n
different integers, the product of all the positive differences of these numbers is divisible by
2014
2014
2014
.
6
1
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Beautiful problem of graph theory.
In a school there are
n
n
n
classes and
n
n
n
students. The students are enrolled in classes, such that no two of them have exactly the same classes. Prove that we can close a class in a such way that there still are no two of them which have exactly the same classes.