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Contests
National and Regional Contests
Mexico Contests
Regional Olympiad of Mexico Center Zone
2020 Regional Olympiad of Mexico Center Zone
2020 Regional Olympiad of Mexico Center Zone
Part of
Regional Olympiad of Mexico Center Zone
Subcontests
(6)
1
1
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red and blue coloring of cells in triangular cells of equilateral triangle
There is a board with the shape of an equilateral triangle with side
n
n
n
divided into triangular cells with the shape of equilateral triangles with side
1
1
1
(the figure below shows the board when
n
=
4
n = 4
n
=
4
). Each and every triangular cell is colored either red or blue. What is the least number of cells that can be colored blue without two red cells sharing one side? https://cdn.artofproblemsolving.com/attachments/0/1/d1f034258966b319dc87297bdb311f134497b5.png
4
1
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ABCD square wanted, 3 circles (O,OA), (A,AO), (P,PQ)
Let
Γ
1
\Gamma_1
Γ
1
be a circle with center
O
O
O
and
A
A
A
a point on it. Consider the circle
Γ
2
\Gamma_2
Γ
2
with center at
A
A
A
and radius
A
O
AO
A
O
. Let
P
P
P
and
Q
Q
Q
be the intersection points of
Γ
1
\Gamma_1
Γ
1
and
Γ
2
\Gamma_2
Γ
2
. Consider the circle
Γ
3
\Gamma_3
Γ
3
with center at
P
P
P
and radius
P
Q
PQ
PQ
. Let
C
C
C
be the second intersection point of
Γ
3
\Gamma_3
Γ
3
and
Γ
1
\Gamma_1
Γ
1
. The line
O
P
OP
OP
cuts
Γ
3
\Gamma_3
Γ
3
at
R
R
R
and
S
S
S
, with
R
R
R
outside
Γ
1
\Gamma_1
Γ
1
.
R
C
RC
RC
cuts
Γ
1
\Gamma_1
Γ
1
into
B
B
B
.
C
S
CS
CS
cuts
Γ
1
\Gamma_1
Γ
1
into
D
D
D
. Show that
A
B
C
D
ABCD
A
BC
D
is a square.
3
1
Hide problems
KB = KC wanted, 2 midpoints, 2 perpendiculars related
In an acute triangle
A
B
C
ABC
A
BC
, an arbitrary point
P
P
P
is chosen on the altitude
A
H
AH
A
H
. The points
E
E
E
and
F
F
F
are the midpoints of
A
C
AC
A
C
and
A
B
AB
A
B
, respectively. The perpendiculars from
E
E
E
on
C
P
CP
CP
and from
F
F
F
on
B
P
BP
BP
intersect at the point
K
K
K
. Show that
K
B
=
K
C
KB = KC
K
B
=
K
C
.
2
1
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2a^2 b^2/(a^5+b^5)+2b^2 c^2/(b^5+c^5)+2c^2 a^2/(c^5+a^5) <= sum (a+b)/(2ab)
Let
a
a
a
,
b
b
b
and
c
c
c
be positive real numbers, prove that
2
a
2
b
2
a
5
+
b
5
+
2
b
2
c
2
b
5
+
c
5
+
2
c
2
a
2
c
5
+
a
5
≤
a
+
b
2
a
b
+
b
+
c
2
b
c
+
c
+
a
2
c
a
\frac{2a^2 b^2}{a^5+b^5}+\frac{2b^2 c^2}{b^5+c^5}+\frac{2c^2 a^2}{c^5+a^5}\le\frac{a+b}{2ab}+\frac{b+c}{2bc}+\frac{c+a}{2ca}
a
5
+
b
5
2
a
2
b
2
+
b
5
+
c
5
2
b
2
c
2
+
c
5
+
a
5
2
c
2
a
2
≤
2
ab
a
+
b
+
2
b
c
b
+
c
+
2
c
a
c
+
a
5
1
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Integers equation
Find all positive integers
m
,
n
m,n
m
,
n
such that
m
2
+
5
n
m^2+5n
m
2
+
5
n
and
n
2
+
5
m
n^2+5m
n
2
+
5
m
are perfect squares.
6
1
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Colouring problem
Let
n
,
k
n,k
n
,
k
be integers such that
n
≥
k
≥
3
n\geq k\geq3
n
≥
k
≥
3
. Consider
n
+
1
n+1
n
+
1
points in a plane (there is no three collinear points) and
k
k
k
different colors, then, we color all the segments that connect every two points. We say that an angle is good if its vertex is one of the initial set, and its two sides aren't the same color. Show that there exist a coloration such that the \\ total number of good angles is greater than
n
(
k
2
)
⌊
(
n
k
)
⌋
2
n \binom{k}{2} \lfloor(\frac{n}{k})\rfloor^2
n
(
2
k
)
⌊(
k
n
)
⌋
2