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Problems
Contests
International Contests
Lusophon Mathematical Olympiad
2019 Lusophon Mathematical Olympiad
2019 Lusophon Mathematical Olympiad
Part of
Lusophon Mathematical Olympiad
Subcontests
(6)
6
1
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piles of stones game, winning strategy
Two players Arnaldo and Betania play alternately, with Arnaldo being the first to play. Initially there are two piles of stones containing
x
x
x
and
y
y
y
stones respectively. In each play, it is possible to perform one of the following operations: 1. Choose two non-empty piles and take one stone from each pile. 2. Choose a pile with an odd amount of stones, take one of their stones and, if possible, split into two piles with the same amount of stones. The player who cannot perform either of operations 1 and 2 loses. Determine who has the winning strategy based on
x
x
x
and
y
y
y
.
3
1
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IG // AB iff CI // IO
Let
A
B
C
ABC
A
BC
be a triangle with
A
C
≠
B
C
AC \ne BC
A
C
=
BC
. In triangle
A
B
C
ABC
A
BC
, let
G
G
G
be the centroid,
I
I
I
the incenter and O Its circumcenter. Prove that
I
G
IG
I
G
is parallel to
A
B
AB
A
B
if, and only if,
C
I
CI
C
I
is perpendicular on
I
O
IO
I
O
.
2
1
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infinite triples a,b,c such a+b +c=n, ax^2 + bx + c = 0 has rational roots
Prove that for every
n
n
n
nonzero integer , there are infinite triples of nonzero integers
a
,
b
a, b
a
,
b
and
c
c
c
that satisfy the conditions: 1.
a
+
b
+
c
=
n
a + b + c = n
a
+
b
+
c
=
n
2.
a
x
2
+
b
x
+
c
=
0
ax^2 + bx + c = 0
a
x
2
+
b
x
+
c
=
0
has rational roots.
1
1
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any 2 consecutive digits of a sequence from 1-9 are divisible by 7 or 13
Find a way to write all the digits of
1
1
1
to
9
9
9
in a sequence and without repetition, so that the numbers determined by any two consecutive digits of the sequence are divisible by
7
7
7
or
13
13
13
.
5
1
Hide problems
2018 = A^5 + B^5 + C^5 + D^5 + E^5, 2018 = A^5 + B^5 + C^5 + D^5
a) Show that there are five integers
A
,
B
,
C
,
D
A, B, C, D
A
,
B
,
C
,
D
, and
E
E
E
such that
2018
=
A
5
+
B
5
+
C
5
+
D
5
+
E
5
2018 = A^5 + B^5 + C^5 + D^5 + E^5
2018
=
A
5
+
B
5
+
C
5
+
D
5
+
E
5
b) Show that there are no four integers
A
,
B
,
C
A, B, C
A
,
B
,
C
and
D
D
D
such that
2018
=
A
5
+
B
5
+
C
5
+
D
5
2018 = A^5 + B^5 + C^5 + D^5
2018
=
A
5
+
B
5
+
C
5
+
D
5
4
1
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2(a^2 + 1)(b^2 + 1) = (a + 1)(b + 1)(ab + 1) in R
Find all the real numbers
a
a
a
and
b
b
b
that satisfy the relation
2
(
a
2
+
1
)
(
b
2
+
1
)
=
(
a
+
1
)
(
b
+
1
)
(
a
b
+
1
)
2(a^2 + 1)(b^2 + 1) = (a + 1)(b + 1)(ab + 1)
2
(
a
2
+
1
)
(
b
2
+
1
)
=
(
a
+
1
)
(
b
+
1
)
(
ab
+
1
)