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Problems
Contests
National and Regional Contests
Mexico Contests
Mexico National Olympiad
2004 Mexico National Olympiad
2004 Mexico National Olympiad
Part of
Mexico National Olympiad
Subcontests
(6)
5
1
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intersection of two lines is the circumcirce of a triangle
Let
ω
1
\omega_1
ω
1
and
ω
2
\omega_2
ω
2
be two circles such that the center
O
O
O
of
ω
2
\omega_2
ω
2
lies in
ω
1
\omega_1
ω
1
. Let
C
C
C
and
D
D
D
be the two intersection points of the circles. Let
A
A
A
be a point on
ω
1
\omega_1
ω
1
and let
B
B
B
be a point on
ω
2
\omega_2
ω
2
such that
A
C
AC
A
C
is tangent to
ω
2
\omega_2
ω
2
in C and BC is tangent to
ω
1
\omega_1
ω
1
in
C
C
C
. The line segment
A
B
AB
A
B
meets
ω
2
\omega_2
ω
2
again in
E
E
E
and also meets
ω
1
\omega_1
ω
1
again in F. The line
C
E
CE
CE
meets
ω
1
\omega_1
ω
1
again in
G
G
G
and the line
C
F
CF
CF
meets the line
G
D
GD
G
D
in
H
H
H
. Prove that the intersection point of
G
O
GO
GO
and
E
H
EH
E
H
is the center of the circumcircle of the triangle
D
E
F
DEF
D
EF
.
6
1
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traveling on the edges of a rectangular array of 2004 x 2004
What is the maximum number of possible change of directions in a path traveling on the edges of a rectangular array of
2004
×
2004
2004 \times 2004
2004
×
2004
, if the path does not cross the same place twice?.
4
1
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How many teams played in the tournament?
At the end of a soccer tournament in which any pair of teams played between them exactly once, and in which there were not draws, it was observed that for any three teams
A
,
B
A, B
A
,
B
and C, if
A
A
A
defeated
B
B
B
and
B
B
B
defeated
C
C
C
, then
A
A
A
defeated
C
C
C
. Any team calculated the difference (positive) between the number of games that it won and the number of games it lost. The sum of all these differences was
5000
5000
5000
. How many teams played in the tournament? Find all possible answers.
3
1
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Mexican equal segments II
Let
Z
Z
Z
and
Y
Y
Y
be the tangency points of the incircle of the triangle
A
B
C
ABC
A
BC
with the sides
A
B
AB
A
B
and
C
A
CA
C
A
, respectively. The parallel line to
Y
Z
Y Z
Y
Z
through the midpoint
M
M
M
of
B
C
BC
BC
, meets
C
A
CA
C
A
in
N
N
N
. Let
L
L
L
be the point in
C
A
CA
C
A
such that
N
L
=
A
B
NL = AB
N
L
=
A
B
(and
L
L
L
on the same side of
N
N
N
than
A
A
A
). The line
M
L
ML
M
L
meets
A
B
AB
A
B
in
K
K
K
. Prove that
K
A
=
N
C
KA = NC
K
A
=
NC
.
2
1
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max No. of positive integers so for that any 2 of them a,b : |a - b| >=ab/100
Find the maximum number of positive integers such that any two of them
a
,
b
a, b
a
,
b
(with
a
≠
b
a \ne b
a
=
b
) satisfy that
∣
a
−
b
∣
≥
a
b
100
.
|a - b| \ge \frac{ab}{100} .
∣
a
−
b
∣
≥
100
ab
.
1
1
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find primes p<q< r so that 25pq + r = 2004 and pqr + 1 is a perfect square
Find all the prime number
p
,
q
p, q
p
,
q
and r with
p
<
q
<
r
p < q < r
p
<
q
<
r
, such that
25
p
q
+
r
=
2004
25pq + r = 2004
25
pq
+
r
=
2004
and
p
q
r
+
1
pqr + 1
pq
r
+
1
is a perfect square.