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Problems
Contests
National and Regional Contests
Mexico Contests
Regional Olympiad of Mexico Center Zone
2011 Regional Olympiad of Mexico Center Zone
2011 Regional Olympiad of Mexico Center Zone
Part of
Regional Olympiad of Mexico Center Zone
Subcontests
(6)
2
1
Hide problems
2 perpendicular lines concurrent with circumcircle
Let
A
B
C
ABC
A
BC
be a triangle and let
L
L
L
,
M
M
M
,
N
N
N
be the midpoints of the sides
B
C
BC
BC
,
C
A
CA
C
A
and
A
B
AB
A
B
, respectively. The points
P
P
P
and
Q
Q
Q
lie on
A
B
AB
A
B
and
B
C
BC
BC
, respectively; the points
R
R
R
and
S
S
S
are such that
N
N
N
is the midpoint of
P
R
PR
PR
and
L
L
L
is the midpoint of
Q
S
QS
QS
. Show that if
P
S
PS
PS
and
Q
R
QR
QR
are perpendicular, then their intersection lies on in the circumcircle of triangle
L
M
N
LMN
L
MN
.
5
1
Hide problems
special partitions of the 100 stones in k piles
There are
100
100
100
stones in a pile. A partition of the heap in
k
k
k
piles is called special if it meets that the number of stones in each pile is different and also for any partition of any of the piles into two new piles it turns out that between the
k
+
1
k + 1
k
+
1
piles there are two that have the same number of stones (each pile contains at least one stone). a) Find the maximum number
k
k
k
, such that there is a special partition of the
100
100
100
stones into
k
k
k
piles. b) Find the minimum number
k
k
k
, such that there is a special partition of the
100
100
100
stones in
k
k
k
piles.
6
1
Hide problems
sum of inradii of circles inscribed in curvilinear triangle <= sum of radius
Given a circle
C
C
C
and a diameter
A
B
AB
A
B
in it, mark a point
P
P
P
on
A
B
AB
A
B
different from the ends. In one of the two arcs determined by
A
B
AB
A
B
choose the points
M
M
M
and
N
N
N
such that
∠
A
P
M
=
6
0
∘
=
∠
B
P
N
\angle APM = 60 ^ \circ = \angle BPN
∠
A
PM
=
6
0
∘
=
∠
BPN
. The segments
M
P
MP
MP
and
N
P
NP
NP
are drawn to obtain three curvilinear triangles;
A
P
M
APM
A
PM
,
M
P
N
MPN
MPN
and
N
P
B
NPB
NPB
(the sides of the curvilinear triangle
A
P
M
APM
A
PM
are the segments
A
P
AP
A
P
and
P
M
PM
PM
and the arc
A
M
AM
A
M
). In each curvilinear triangle a circle is inscribed, that is, a circle is built tangent to the three sides. Show that the sum of the radii of the three inscribed circles is less than or equal to the radius of
C
C
C
.
3
1
Hide problems
max of n positive integers <10000, not prime, but any 2 coprime
We have
n
n
n
positive integers greater than
1
1
1
and less than
10000
10000
10000
such that neither of them is prime but any two of them are relative prime. Find the maximum value of
n
n
n
.
4
1
Hide problems
property of a 6n digit number divisible by 7
Show that if a
6
n
6n
6
n
-digit number is divisible by
7
7
7
, then the number that results from moving the ones digit to the beginning of the number is also a multiple of
7
7
7
.
1
1
Hide problems
8 people around a circular table with coins
Eight people are sitting at a circular table, it is known that any three consecutive people at the table have an odd number of coins (among the three people), show that each person has at least one coin.