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Problems
Contests
International Contests
Lusophon Mathematical Olympiad
2024 Lusophon Mathematical Olympiad
2024 Lusophon Mathematical Olympiad
Part of
Lusophon Mathematical Olympiad
Subcontests
(6)
6
1
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Oeirense numbers
A positive integer
n
n
n
is called
o
e
i
r
e
n
s
e
oeirense
oe
i
re
n
se
if there exist two positive integers
a
a
a
and
b
b
b
, not necessarily distinct, such that
n
=
a
2
+
b
2
n=a^2+b^2
n
=
a
2
+
b
2
.Determine the greatest integer
k
k
k
such that there exist infinitely many positive integers
n
n
n
such that
n
n
n
,
n
+
1
n+1
n
+
1
,
…
\dots
…
,
n
+
k
n+k
n
+
k
are oeirenses.
5
1
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How many ways to paint the board?
In a
9
×
9
9\times9
9
×
9
board, the squares are labeled from 11 to 99, with the first digit indicating the row and the second digit indicating the column. One would like to paint the squares in black or white in a way that each black square is adjacent to at most one other black square and each white square is adjacent to at most one other white square. Two squares are adjacent if they share a common side. How many ways are there to paint the board such that the squares
44
44
44
and
49
49
49
are both black?
3
1
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Prove that the circumcentre appears on a circumcircle
Let
A
B
C
ABC
A
BC
be a triangle with incentre
I
I
I
. A line
r
r
r
that passes through
I
I
I
intersects the circumcircles of triangles
A
I
B
AIB
A
I
B
and
A
I
C
AIC
A
I
C
at points
P
P
P
and
Q
Q
Q
, respectively. Prove that the circumcentre of triangle
A
P
Q
APQ
A
PQ
is on the circumcircle of
A
B
C
ABC
A
BC
.
2
1
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Find the greatest n that always divides the product
For each set of five integers
S
=
{
a
1
,
a
2
,
a
3
,
a
4
,
a
5
}
S= \{a_1, a_2, a_3, a_4, a_5\}
S
=
{
a
1
,
a
2
,
a
3
,
a
4
,
a
5
}
, let
P
S
P_S
P
S
be the product of all differences between two of the elements, namely
P
S
=
(
a
5
−
a
1
)
(
a
4
−
a
1
)
(
a
3
−
a
1
)
(
a
2
−
a
1
)
(
a
5
−
a
2
)
(
a
4
−
a
2
)
(
a
3
−
a
2
)
(
a
5
−
a
3
)
(
a
4
−
a
3
)
(
a
5
−
a
4
)
P_S=(a_5-a_1)(a_4-a_1)(a_3-a_1)(a_2-a_1)(a_5-a_2)(a_4-a_2)(a_3-a_2)(a_5-a_3)(a_4-a_3)(a_5-a_4)
P
S
=
(
a
5
−
a
1
)
(
a
4
−
a
1
)
(
a
3
−
a
1
)
(
a
2
−
a
1
)
(
a
5
−
a
2
)
(
a
4
−
a
2
)
(
a
3
−
a
2
)
(
a
5
−
a
3
)
(
a
4
−
a
3
)
(
a
5
−
a
4
)
Determine the greatest integer
n
n
n
such that given any set
S
S
S
of five integers,
n
n
n
divides
P
S
P_S
P
S
.
1
1
Hide problems
Find geometric progressions.
Determine all geometric progressions such that the product of the three first terms is
64
64
64
and the sum of them is
14
14
14
.
4
1
Hide problems
Find the shaded area
In the figure, the triangles
A
B
C
ABC
A
BC
and
C
D
E
CDE
C
D
E
are equilateral, with side lengths
1
1
1
and
4
4
4
, respectively. Moreover,
B
B
B
,
C
C
C
and
D
D
D
are collinear and
F
F
F
and
G
G
G
are midpoints of
B
C
BC
BC
and
C
D
CD
C
D
, respectively. Let
P
P
P
be the intersection point of
A
F
AF
A
F
and
B
E
BE
BE
.Determine the area of the shaded triangle
B
P
G
BPG
BPG
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