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Contests
National and Regional Contests
Mexico Contests
Regional Olympiad of Mexico Center Zone
2016 Regional Olympiad of Mexico Center Zone
2016 Regional Olympiad of Mexico Center Zone
Part of
Regional Olympiad of Mexico Center Zone
Subcontests
(6)
6
1
Hide problems
you can get from 1 to another by taking buses that only pass through cities in
In Tlaxcala, there is a transportation system that works through buses that travel from one city to another in one direction . A set
S
S
S
of cities is said beautiful if it contains at least three different cities and from each city
A
A
A
in
S
S
S
at least two buses depart, each one goes directly to a different city in
S
S
S
and none of them is
A
A
A
(if there is a direct bus from
A
A
A
to a city
B
B
B
in
S
S
S
, there is not necessarily a direct bus from
B
B
B
to
A
A
A
). Show that if there exists a beautiful set of cities
S
S
S
, then there exists a beautiful
T
T
T
subset of
S
S
S
, such that for any two cities in
T
T
T
, you can get from one to another by taking buses that only pass through cities in
T
T
T
. Note: A bus goes directly from one city to another if it does not pass through any other city.
5
1
Hide problems
is there a maximal tlaxcalteca arithmetic progression of 11 elements?
An arithmetic sequence is a sequence of
(
a
1
,
a
2
,
…
,
a
n
)
(a_1, a_2, \dots, a_n)
(
a
1
,
a
2
,
…
,
a
n
)
such that the difference between any two consecutive terms is the same. That is,
a
i
+
1
−
a
i
=
d
a_ {i + 1} -a_i = d
a
i
+
1
−
a
i
=
d
for all
i
∈
{
1
,
2
,
…
,
n
−
1
}
i \in \{1,2, \dots, n-1 \}
i
∈
{
1
,
2
,
…
,
n
−
1
}
, where
d
d
d
is the difference of the progression. A sequence
(
a
1
,
a
2
,
…
,
a
n
)
(a_1, a_2, \dots, a_n)
(
a
1
,
a
2
,
…
,
a
n
)
is tlaxcalteca if for all
i
∈
{
1
,
2
,
…
,
n
−
1
}
i \in \{1,2, \dots, n-1 \}
i
∈
{
1
,
2
,
…
,
n
−
1
}
, there exists
m
i
m_i
m
i
positive integer such that
a
i
=
1
m
i
a_i = \frac {1} {m_i}
a
i
=
m
i
1
. A taxcalteca arithmetic progression
(
a
1
,
a
2
,
…
,
a
n
)
(a_1, a_2, \dots, a_n )
(
a
1
,
a
2
,
…
,
a
n
)
is said to be maximal if
(
a
1
−
d
,
a
1
,
a
2
,
…
,
a
n
)
(a_1-d, a_1, a_2, \dots, a_n)
(
a
1
−
d
,
a
1
,
a
2
,
…
,
a
n
)
and
(
a
1
,
a
2
,
…
,
a
n
,
a
n
+
d
)
(a_1, a_2, \dots, a_n, a_n + d)
(
a
1
,
a
2
,
…
,
a
n
,
a
n
+
d
)
are not Tlaxcalan arithmetic progressions. Is there a maximal tlaxcalteca arithmetic progression of
11
11
11
elements?
4
1
Hide problems
vertex of # AMPN is circumcenter of ABC, starting with 2 intersecting circles
Let
A
A
A
be one of the two points where the circles whose centers are the points
M
M
M
and
N
N
N
intersect. The tangents in
A
A
A
to such circles intersect them again in
B
B
B
and
C
C
C
, respectively. Let
P
P
P
be a point such that the quadrilateral
A
M
P
N
AMPN
A
MPN
is a parallelogram. Show that
P
P
P
is the circumcenter of triangle
A
B
C
ABC
A
BC
.
3
1
Hide problems
2 lines intersect on circumcircle, orthocenter , reflections of line wrt lines
Let
A
B
C
ABC
A
BC
be a triangle with orthocenter
H
H
H
and
ℓ
\ell
ℓ
a line that passes through
H
H
H
, and is parallel to
B
C
BC
BC
. Let
m
m
m
and
n
n
n
be the reflections of
ℓ
\ell
ℓ
on the sides of
A
B
AB
A
B
and
A
C
AC
A
C
, respectively,
m
m
m
and
n
n
n
are intersect at
P
P
P
. If
H
P
HP
H
P
and
B
C
BC
BC
intersect at
Q
Q
Q
, prove that the parallel to
A
H
AH
A
H
through
Q
Q
Q
and
A
P
AP
A
P
intersect at the circumcenter of the triangle
A
B
C
ABC
A
BC
.
2
1
Hide problems
2 player game 7 piles of 2014 pebbles and 1 of 2008 pebbles
There are seven piles with
2014
2014
2014
pebbles each and a pile with
2008
2008
2008
pebbles. Ana and Beto play in turns and Ana always plays first. One move consists of removing pebbles from all the piles. From each pile is removed a different amount of pebbles, between
1
1
1
and
8
8
8
pebbles. The first player who cannot make a move loses. a) Who has a winning strategy? b) If there were seven piles with
2015
2015
2015
pebbles each and a pile with
2008
2008
2008
pebbles, who has a winning strategy?
1
1
Hide problems
ways to fill x 3x8 grid by 9 numbers having 6 already in fixed places
The grid shown below is completed by choosing nine of the following numbers without repeating:
4
,
5
,
6
,
7
,
8
,
12
,
13
,
16
,
18
,
19
4, 5, 6, 7, 8, 12, 13, 16, 18, 19
4
,
5
,
6
,
7
,
8
,
12
,
13
,
16
,
18
,
19
. If the sum of the five rows are equal to each other and the sum of the three columns are equal to each other, in how many different ways is it possible to fill the grid?
10
9
3
11
17
20
\begin {array} {| c | c | c |} \hline 10 & & \\ \hline & & 9 \\ \hline & 3 & \\ \hline 11 & & 17 \\ \hline & 20 & \\ \hline \end {array}
10
11
3
20
9
17
Note: The sum of the rows and the sum of the columns are not necessarily equal.