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International Contests
Lusophon Mathematical Olympiad
2013 Lusophon Mathematical Olympiad
2013 Lusophon Mathematical Olympiad
Part of
Lusophon Mathematical Olympiad
Subcontests
(6)
6
1
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III Lusophon Mathematical Olympiad 2013 - Problem 6
Consider a triangle
A
B
C
ABC
A
BC
. Let
S
S
S
be a circumference in the interior of the triangle that is tangent to the sides
B
C
BC
BC
,
C
A
CA
C
A
,
A
B
AB
A
B
at the points
D
D
D
,
E
E
E
,
F
F
F
respectively. In the exterior of the triangle we draw three circumferences
S
A
S_A
S
A
,
S
B
S_B
S
B
,
S
C
S_C
S
C
. The circumference
S
A
S_A
S
A
is tangent to
B
C
BC
BC
at
L
L
L
and to the prolongation of the lines
A
B
AB
A
B
,
A
C
AC
A
C
at the points
M
M
M
,
N
N
N
respectively. The circumference
S
B
S_B
S
B
is tangent to
A
C
AC
A
C
at
E
E
E
and to the prolongation of the line
B
C
BC
BC
at
P
P
P
. The circumference
S
C
S_C
S
C
is tangent to
A
B
AB
A
B
at
F
F
F
and to the prolongation of the line
B
C
BC
BC
at
Q
Q
Q
. Show that the lines
E
P
EP
EP
,
F
Q
FQ
FQ
and
A
L
AL
A
L
meet at a point of the circumference
S
S
S
.
5
1
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III Lusophon Mathematical Olympiad 2013 - Problem 5
Find all the numbers of
5
5
5
non-zero digits such that deleting consecutively the digit of the left, in each step, we obtain a divisor of the previous number.
4
1
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III Lusophon Mathematical Olympiad 2013 - Problem 4
Find all the pairs
(
x
,
y
)
(x,y)
(
x
,
y
)
of positive integers that satisfy the equation
x
2
−
x
y
+
2
x
−
3
y
=
2013
x^2-xy+2x-3y=2013
x
2
−
x
y
+
2
x
−
3
y
=
2013
.
3
1
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III Lusophon Mathematical Olympiad 2013 - Problem 3
An event occurs many years ago. It occurs periodically in
x
x
x
consecutive years, then there is a break of
y
y
y
consecutive years. We know that the event occured in
1964
1964
1964
,
1986
1986
1986
,
1996
1996
1996
,
2008
2008
2008
and it didn't occur in
1976
1976
1976
,
1993
1993
1993
,
2006
2006
2006
,
2013
2013
2013
. What is the first year in that the event will occur again?
2
1
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III Lusophon Mathematical Olympiad 2013 - Problem 2
Let
A
B
C
ABC
A
BC
be an acute triangle. The circumference with diameter
A
B
AB
A
B
intersects sides
A
C
AC
A
C
and
B
C
BC
BC
at
E
E
E
and
F
F
F
respectively. The tangent lines to the circumference at the points
E
E
E
and
F
F
F
meet at
P
P
P
. Show that
P
P
P
belongs to the altitude from
C
C
C
of triangle
A
B
C
ABC
A
BC
.
1
1
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III Lusophon Mathematical Olympiad 2013 - Problem 1
If Xiluva puts two oranges in each basket, four oranges are in excess. If she puts five oranges in each basket, one basket is in excess. How many oranges and baskets has Xiluva?