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Problems
Contests
National and Regional Contests
Mexico Contests
Regional Olympiad of Mexico Center Zone
2008 Regional Olympiad of Mexico Center Zone
2008 Regional Olympiad of Mexico Center Zone
Part of
Regional Olympiad of Mexico Center Zone
Subcontests
(6)
6
1
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AP _|_BQ if AQ_|_DP, AB = AD, <B = < D = 90 ^o
In the quadrilateral
A
B
C
D
ABCD
A
BC
D
, we have
A
B
=
A
D
AB = AD
A
B
=
A
D
and
∠
B
=
∠
D
=
9
0
∘
\angle B = \angle D = 90 ^ \circ
∠
B
=
∠
D
=
9
0
∘
. The points
P
P
P
and
Q
Q
Q
lie on
B
C
BC
BC
and
C
D
CD
C
D
, respectively, so that
A
Q
AQ
A
Q
is perpendicular on
D
P
DP
D
P
. Prove that
A
P
AP
A
P
is perpendicular to
B
Q
BQ
BQ
.
5
1
Hide problems
max prime that divides S = p_1 + p_2 + p_3 + ...+ p_ {999}, product of digits
Each positive integer number
n
g
e
1
n \ ge 1
n
g
e
1
is assigned the number
p
n
p_n
p
n
which is the product of all its non-zero digits. For example,
p
6
=
6
p_6 = 6
p
6
=
6
,
p
32
=
6
p_ {32} = 6
p
32
=
6
,
p
203
=
6
p_ {203} = 6
p
203
=
6
. Let
S
=
p
1
+
p
2
+
p
3
+
⋯
+
p
999
S = p_1 + p_2 + p_3 + \dots + p_ {999}
S
=
p
1
+
p
2
+
p
3
+
⋯
+
p
999
. Find the largest prime that divides
S
S
S
.
4
1
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min no of colors for n points and their segments, under 2 conditions
Let
n
n
n
points, where there are not
3
3
3
of them on a line, and consider the segments that are formed by connecting any
2
2
2
of the points. There are enough colors available to paint the points and the segments, coloring them with the following two rules: a) All the segments that reach the same point are painted of different colors. b) Each point is painted a different color to all the segments that reach it. Find the minimum number of colors needed to make such a coloring.
3
1
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each unit square has two red and two blue vertices in nxn grid
Consider a
n
×
n
n \times n
n
×
n
grid divided into
n
2
n ^ 2
n
2
squares of
1
×
1
1 \times 1
1
×
1
. Each of the
(
n
+
1
)
2
(n + 1) ^ 2
(
n
+
1
)
2
vertices of the grid is colored red or blue. Find the number of coloring such that each unit square has two red and two blue vertices.
2
1
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BLI , L'IB are similar iff AC = AB + BL , incenter, circumcircle related
Let
A
B
C
ABC
A
BC
be a triangle with incenter
I
I
I
, the line
A
I
AI
A
I
intersects
B
C
BC
BC
at
L
L
L
and the circumcircle of
A
B
C
ABC
A
BC
at
L
′
L'
L
′
. Show that the triangles
B
L
I
BLI
B
L
I
and
L
′
I
B
L'IB
L
′
I
B
are similar if and only if
A
C
=
A
B
+
B
L
AC = AB + BL
A
C
=
A
B
+
B
L
.
1
1
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a ^2 - 3a = b ^3 - 2 diophantine
Find all pairs of integers
a
,
b
a, b
a
,
b
that satisfy
a
2
−
3
a
=
b
3
−
2
a ^2-3a = b ^3-2
a
2
−
3
a
=
b
3
−
2
.