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Contests
International Contests
Lusophon Mathematical Olympiad
2018 Lusophon Mathematical Olympiad
2018 Lusophon Mathematical Olympiad
Part of
Lusophon Mathematical Olympiad
Subcontests
(6)
2
1
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given right & isosceles triange, another isosceles, 2 midpoints, square wanted
In a triangle
A
B
C
ABC
A
BC
, right in
A
A
A
and isosceles, let
D
D
D
be a point on the side
A
C
AC
A
C
(
A
≠
D
≠
C
A \ne D \ne C
A
=
D
=
C
) and
E
E
E
be the point on the extension of
B
A
BA
B
A
such that the triangle
A
D
E
ADE
A
D
E
is isosceles. Let
P
P
P
be the midpoint of segment
B
D
BD
B
D
,
R
R
R
be the midpoint of the segment
C
E
CE
CE
and
Q
Q
Q
the intersection point of
E
D
ED
E
D
and
B
C
BC
BC
. Prove that the quadrilateral
A
R
Q
P
ARQP
A
RQP
is a square.
1
1
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numbers fill so that 5 lines have constant sum and four corners have sum 123
Fill in the corners of the square, so that the sum of the numbers in each one of the
5
5
5
lines of the square is the same and the sum of the four corners is
123
123
123
.
6
1
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1 x 3 pieces in a 3 x 25 board,m max no of pieces placed
In a
3
×
25
3 \times 25
3
×
25
board,
1
×
3
1 \times 3
1
×
3
pieces are placed (vertically or horizontally) so that they occupy entirely
3
3
3
boxes on the board and do not have a common point. What is the maximum number of pieces that can be placed, and for that number, how many configurations are there?[hide=original formulation] Num tabuleiro 3 × 25 s˜ao colocadas pe¸cas 1 × 3 (na vertical ou na horizontal) de modo que ocupem inteiramente 3 casas do tabuleiro e n˜ao se toquem em nenhum ponto. Qual ´e o n´umero m´aximo de pe¸cas que podem ser colocadas, e para esse n´umero, quantas configura¸c˜oes existem? [url=https://www.obm.org.br/content/uploads/2018/09/Provas_OMCPLP_2018.pdf]source
5
1
Hide problems
geometric sequence with 3 integer terms of sum 57
Determine the increasing geometric progressions, with three integer terms, such that the sum of these terms is
57
57
57
4
1
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m^2=n^2 +m+n+2018 , solve in positive integers
Determine the pairs of positive integer numbers
m
m
m
and
n
n
n
that satisfy the equation
m
2
=
n
2
+
m
+
n
+
2018
m^2=n^2 +m+n+2018
m
2
=
n
2
+
m
+
n
+
2018
.
3
1
Hide problems
smallest a so that S(n)-S(n+a) = 2018, where S(n)=sum of digits
For each positive integer
n
n
n
, let
S
(
n
)
S(n)
S
(
n
)
be the sum of the digits of
n
n
n
. Determines the smallest positive integer
a
a
a
such that there are infinite positive integers
n
n
n
for which you have
S
(
n
)
−
S
(
n
+
a
)
=
2018
S (n) -S (n + a) = 2018
S
(
n
)
−
S
(
n
+
a
)
=
2018
.