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Problems
Contests
National and Regional Contests
Mexico Contests
Mexico National Olympiad
2005 Mexico National Olympiad
2005 Mexico National Olympiad
Part of
Mexico National Olympiad
Subcontests
(6)
6
1
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Mexican equal segments
Let
A
B
C
ABC
A
BC
be a triangle and
A
D
AD
A
D
be the angle bisector of
<
B
A
C
<BAC
<
B
A
C
, with
D
D
D
on
B
C
BC
BC
. Let
E
E
E
be a point on segment
B
C
BC
BC
such that
B
D
=
E
C
BD = EC
B
D
=
EC
. Through
E
E
E
draw
l
l
l
a parallel line to
A
D
AD
A
D
and let
P
P
P
be a point in
l
l
l
inside the triangle. Let
G
G
G
be the point where
B
P
BP
BP
intersects
A
C
AC
A
C
and
F
F
F
be the point where
C
P
CP
CP
intersects
A
B
AB
A
B
. Show
B
F
=
C
G
BF = CG
BF
=
CG
.
5
1
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deck with N^3 cards, each card has 1 of N colors, figures and numbers
Let
N
N
N
be an integer greater than
1
1
1
. A deck has
N
3
N^3
N
3
cards, each card has one of
N
N
N
colors, has one of
N
N
N
figures and has one of
N
N
N
numbers (there are no two identical cards). A collection of cards of the deck is "complete" if it has cards of every color, or if it has cards of every figure or of all numbers. How many non-complete collections are there such that, if you add any other card from the deck, the collection becomes complete?
4
1
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an arithmetic trio a_i, a_j, a_k if i < j < k and 2a_j = a_i + a_k.
A list of numbers
a
1
,
a
2
,
…
,
a
m
a_1,a_2,\ldots,a_m
a
1
,
a
2
,
…
,
a
m
contains an arithmetic trio
a
i
,
a
j
,
a
k
a_i, a_j, a_k
a
i
,
a
j
,
a
k
if
i
<
j
<
k
i < j < k
i
<
j
<
k
and
2
a
j
=
a
i
+
a
k
2a_j = a_i + a_k
2
a
j
=
a
i
+
a
k
. Let
n
n
n
be a positive integer. Show that the numbers
1
,
2
,
3
,
…
,
n
1, 2, 3, \ldots, n
1
,
2
,
3
,
…
,
n
can be reordered in a list that does not contain arithmetic trios.
2
1
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matrices with integers and difference between 2 adjacent entries <=N
Given several matrices of the same size. Given a positive integer
N
N
N
, let's say that a matrix is
N
N
N
-balanced if the entries of the matrix are integers and the difference between any two adjacent entries of the matrix is less than or equal to
N
N
N
. (i) Show that every
2
N
2N
2
N
-balanced matrix can be written as a sum of two
N
N
N
-balanced matrices. (ii) Show that every
3
N
3N
3
N
-balanced matrix can be written as a sum of three
N
N
N
-balanced matrices.
1
1
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3 equal circumcircles
Let
O
O
O
be the center of the circumcircle of an acute triangle
A
B
C
ABC
A
BC
, let
P
P
P
be any point inside the segment
B
C
BC
BC
. Suppose the circumcircle of triangle
B
P
O
BPO
BPO
intersects the segment
A
B
AB
A
B
at point
R
R
R
and the circumcircle of triangle
C
O
P
COP
COP
intersects
C
A
CA
C
A
at point
Q
Q
Q
. (i) Consider the triangle
P
Q
R
PQR
PQR
, show that it is similar to triangle
A
B
C
ABC
A
BC
and that
O
O
O
is its orthocenter. (ii) Show that the circumcircles of triangles
B
P
O
BPO
BPO
,
C
O
P
COP
COP
,
P
Q
R
PQR
PQR
have the same radius.
3
1
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Problem 3
Already the complete problem: Determine all pairs
(
a
,
b
)
(a,b)
(
a
,
b
)
of integers different from
0
0
0
for which it is possible to find a positive integer
x
x
x
and an integer
y
y
y
such that
x
x
x
is relatively prime to
b
b
b
and in the following list there is an infinity of integers:
→
a
+
x
y
b
\rightarrow\qquad\frac{a + xy}{b}
→
b
a
+
x
y
,
a
+
x
y
2
b
2
\frac{a + xy^2}{b^2}
b
2
a
+
x
y
2
,
a
+
x
y
3
b
3
\frac{a + xy^3}{b^3}
b
3
a
+
x
y
3
,
…
\ldots
…
,
a
+
x
y
n
b
n
\frac{a + xy^n}{b^n}
b
n
a
+
x
y
n
,
…
\ldots
…
One idea? :arrow: [url=http://www.mathlinks.ro/Forum/viewtopic.php?t=61319]View all the problems from XIX Mexican Mathematical Olympiad