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Contests
International Contests
Lusophon Mathematical Olympiad
2023 Lusophon Mathematical Olympiad
2023 Lusophon Mathematical Olympiad
Part of
Lusophon Mathematical Olympiad
Subcontests
(6)
4
1
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Determine lusophone numbers
A positive integer with 3 digits
A
B
C
‾
\overline{ABC}
A
BC
is
L
u
s
o
p
h
o
n
Lusophon
Lu
so
p
h
o
n
if
A
B
C
‾
+
C
B
A
‾
\overline{ABC}+\overline{CBA}
A
BC
+
CB
A
is a perfect square. Find all
L
u
s
o
p
h
o
n
Lusophon
Lu
so
p
h
o
n
numbers.
6
1
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Calculator with 2 operations, possible to make r
A calculator has two operations
A
A
A
and
B
B
B
and initially shows the number
1
1
1
. Operation
A
A
A
turns
x
x
x
into
x
+
1
x+1
x
+
1
and operation B turns
x
x
x
into
x
x
+
1
\dfrac{x}{x+1}
x
+
1
x
. a) Show all the ways we can get the number
20
23
\dfrac{20}{23}
23
20
. b) For every rational
r
≠
1
r \neq 1
r
=
1
, determine if it is possible to get
r
r
r
using only operations
A
A
A
and
B
B
B
.
5
1
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Regular hexagon, determine XAY
Let
A
B
C
D
E
F
ABCDEF
A
BC
D
EF
be a regular hexagon with side 1. Point
X
,
Y
X, Y
X
,
Y
are on sides
C
D
CD
C
D
and
D
E
DE
D
E
respectively, such that the perimeter of
D
X
Y
DXY
D
X
Y
is
2
2
2
. Determine
∠
X
A
Y
\angle XAY
∠
X
A
Y
.
3
1
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k-special numbers
An integer
n
n
n
is called
k
k
k
-special, with
k
k
k
a positive integer, if it's the sum of the squares of
k
k
k
consecutive integers. For example,
13
13
13
is
2
2
2
-special, since
13
=
2
2
+
3
2
13=2^2+3^2
13
=
2
2
+
3
2
, and
2
2
2
is
3
3
3
-special, since
2
=
(
−
1
)
2
+
0
2
+
1
2
2=(-1)^2+0^2+1^2
2
=
(
−
1
)
2
+
0
2
+
1
2
.a) Prove that there's no perfect square that is
4
4
4
-special.b) Find a perfect square that is
I
2
I^2
I
2
-special, for some odd positive integer
I
I
I
with
I
≥
3
I\ge 3
I
≥
3
.
2
1
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Solve for the angle
Let
D
D
D
be a point on the inside of triangle
A
B
C
ABC
A
BC
such that
A
D
=
C
D
AD=CD
A
D
=
C
D
,
∠
D
A
B
=
7
0
∘
\angle DAB=70^{\circ}
∠
D
A
B
=
7
0
∘
,
∠
D
B
A
=
3
0
∘
\angle DBA=30^{\circ}
∠
D
B
A
=
3
0
∘
and
∠
D
B
C
=
2
0
∘
\angle DBC=20^{\circ}
∠
D
BC
=
2
0
∘
. Find the measure of angle
∠
D
C
B
\angle DCB
∠
D
CB
.
1
1
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Overcomplicated wording for a simple combinatorics problem
A long time ago, there existed Martians with
3
3
3
different colours: red, green and blue. As Mars was devastated by an intergalactic war, only
2
2
2
Martians of each colours survived. In order to reconstruct the Martian population, they decided to use a machine that transforms two Martians of distinct colours into four Martians of colour different to the two initial ones. For example, if a red Martian and a blue Martian use the machine, they'll be transformed into four green Martians.a) Is it possible that, after using that machine finitely many times, we have
2022
2022
2022
red Martians,
2022
2022
2022
green Martians and
2022
2022
2022
blue Martians?b) Is it possible that, after using that machine finitely many times, we have
2021
2021
2021
red Martians,
2022
2022
2022
green Martians and
2023
2023
2023
blue Martians?