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Problems
Contests
International Contests
Lusophon Mathematical Olympiad
2021 Lusophon Mathematical Olympiad
2021 Lusophon Mathematical Olympiad
Part of
Lusophon Mathematical Olympiad
Subcontests
(6)
6
1
Hide problems
Omopeiro numbers
A positive integer
n
n
n
is called
o
m
o
p
e
i
r
o
omopeiro
o
m
o
p
e
i
ro
if there exists
n
n
n
non-zero integers that are not necessarily distinct such that
2021
2021
2021
is the sum of the squares of those
n
n
n
integers. For example, the number
2
2
2
is not an
o
m
o
p
e
i
r
o
omopeiro
o
m
o
p
e
i
ro
, because
2021
2021
2021
is not a sum of two non-zero squares, but
2021
2021
2021
is an
o
m
o
p
e
i
r
o
omopeiro
o
m
o
p
e
i
ro
, because
2021
=
1
2
+
1
2
+
⋯
+
1
2
2021=1^2+1^2+ \dots +1^2
2021
=
1
2
+
1
2
+
⋯
+
1
2
, which is a sum of
2021
2021
2021
squares of the number
1
1
1
.Prove that there exist more than 1500
o
m
o
p
e
i
r
o
omopeiro
o
m
o
p
e
i
ro
numbers.Note: proving that there exist at least 500
o
m
o
p
e
i
r
o
omopeiro
o
m
o
p
e
i
ro
numbers is worth 2 points.
5
1
Hide problems
Four inscribed circles
There are 3 lines
r
,
s
r, s
r
,
s
and
t
t
t
on a plane. The lines
r
r
r
and
s
s
s
intersect perpendicularly at point
A
A
A
. the line
t
t
t
intersects the line
r
r
r
at point
B
B
B
and the line
s
s
s
at point
C
C
C
. There exist exactly 4 circumferences on the plane that are simultaneously tangent to all those 3 lines.Prove that the radius of one of those circumferences is equal to the sum of the radius of the other three circumferences.
4
1
Hide problems
Cyclic equality
Let
x
1
,
x
2
,
x
3
,
x
4
,
x
5
∈
R
+
x_1, x_2, x_3, x_4, x_5\in\mathbb{R}^+
x
1
,
x
2
,
x
3
,
x
4
,
x
5
∈
R
+
such that
x
1
2
−
x
1
x
2
+
x
2
2
=
x
2
2
−
x
2
x
3
+
x
3
2
=
x
3
2
−
x
3
x
4
+
x
4
2
=
x
4
2
−
x
4
x
5
+
x
5
2
=
x
5
2
−
x
5
x
1
+
x
1
2
x_1^2-x_1x_2+x_2^2=x_2^2-x_2x_3+x_3^2=x_3^2-x_3x_4+x_4^2=x_4^2-x_4x_5+x_5^2=x_5^2-x_5x_1+x_1^2
x
1
2
−
x
1
x
2
+
x
2
2
=
x
2
2
−
x
2
x
3
+
x
3
2
=
x
3
2
−
x
3
x
4
+
x
4
2
=
x
4
2
−
x
4
x
5
+
x
5
2
=
x
5
2
−
x
5
x
1
+
x
1
2
Prove that
x
1
=
x
2
=
x
3
=
x
4
=
x
5
x_1=x_2=x_3=x_4=x_5
x
1
=
x
2
=
x
3
=
x
4
=
x
5
.
3
1
Hide problems
A line tangent to a circumcircle
Let triangle
A
B
C
ABC
A
BC
be an acute triangle with
A
B
≠
A
C
AB\neq AC
A
B
=
A
C
. The bisector of
B
C
BC
BC
intersects the lines
A
B
AB
A
B
and
A
C
AC
A
C
at points
F
F
F
and
E
E
E
, respectively. The circumcircle of triangle
A
E
F
AEF
A
EF
has center
P
P
P
and intersects the circumcircle of triangle
A
B
C
ABC
A
BC
at point
D
D
D
with
D
D
D
different to
A
A
A
.Prove that the line
P
D
PD
P
D
is tangent to the circumcircle of triangle
A
B
C
ABC
A
BC
.
2
1
Hide problems
Weird knight move
Esmeralda has created a special knight to play on quadrilateral boards that are identical to chessboards. If a knight is in a square then it can move to another square by moving 1 square in one direction and 3 squares in a perpendicular direction (which is a diagonal of a
2
×
4
2\times4
2
×
4
rectangle instead of
2
×
3
2\times3
2
×
3
like in chess). In this movement, it doesn't land on the squares between the beginning square and the final square it lands on.A trip of the length
n
n
n
of the knight is a sequence of
n
n
n
squares
C
1
,
C
2
,
.
.
.
,
C
n
C1, C2, ..., Cn
C
1
,
C
2
,
...
,
C
n
which are all distinct such that the knight starts at the
C
1
C1
C
1
square and for each
i
i
i
from
1
1
1
to
n
−
1
n-1
n
−
1
it can use the movement described before to go from the
C
i
Ci
C
i
square to the
C
(
i
+
1
)
C(i+1)
C
(
i
+
1
)
.Determine the greatest
N
∈
N
N \in \mathbb{N}
N
∈
N
such that there exists a path of the knight with length
N
N
N
on a
5
×
5
5\times5
5
×
5
board.
1
1
Hide problems
Sextalternado numbers
Juca has decided to call all positive integers with 8 digits as
s
e
x
t
a
l
t
e
r
n
a
d
o
s
sextalternados
se
x
t
a
lt
er
na
d
os
if it is a multiple of 30 and its consecutive digits have different parity. At the same time, Carlos decided to classify all
s
e
x
t
a
l
t
e
r
n
a
d
o
s
sextalternados
se
x
t
a
lt
er
na
d
os
that are multiples of 12 as
s
u
p
e
r
s
e
x
t
a
l
t
e
r
n
a
d
o
s
super sextalternados
s
u
p
erse
x
t
a
lt
er
na
d
os
.a) Show that
s
u
p
e
r
s
e
x
t
a
l
t
e
r
n
a
d
o
s
super sextalternados
s
u
p
erse
x
t
a
lt
er
na
d
os
numbers don't exist.b) Find the smallest
s
e
x
t
a
l
t
e
r
n
a
d
o
sextalternado
se
x
t
a
lt
er
na
d
o
number.