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Problems
Contests
International Contests
Lusophon Mathematical Olympiad
2020 Lusophon Mathematical Olympiad
2020 Lusophon Mathematical Olympiad
Part of
Lusophon Mathematical Olympiad
Subcontests
(6)
6
1
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Floor function for all n
Prove that
⌊
9
n
+
7
⌋
=
⌊
n
+
n
+
1
+
n
+
2
⌋
\lfloor{\sqrt{9n+7}}\rfloor=\lfloor{\sqrt{n}+\sqrt{n+1}+\sqrt{n+2}}\rfloor
⌊
9
n
+
7
⌋
=
⌊
n
+
n
+
1
+
n
+
2
⌋
for all postive integer
n
n
n
.
5
1
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4x4 with products equal 2020
In how many ways can we fill the cells of a
4
×
4
4\times4
4
×
4
grid such that each cell contains exactly one positive integer and the product of the numbers in each row and each column is
2020
2020
2020
?
4
1
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Incircle, circumcenters and similar triangles
Let
A
B
C
ABC
A
BC
be an acute triangle. Its incircle touches the sides
B
C
BC
BC
,
C
A
CA
C
A
and
A
B
AB
A
B
at the points
D
D
D
,
E
E
E
and
F
F
F
, respectively. Let
P
P
P
,
Q
Q
Q
and
R
R
R
be the circumcenters of triangles
A
E
F
AEF
A
EF
,
B
D
F
BDF
B
D
F
and
C
D
E
CDE
C
D
E
, respectively. Prove that triangles
A
B
C
ABC
A
BC
and
P
Q
R
PQR
PQR
are similar.
3
1
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Geometric inequality
Let
A
B
C
ABC
A
BC
be a triangle and on the sides we draw, externally, the squares
B
A
D
E
,
C
B
F
G
BADE, CBFG
B
A
D
E
,
CBFG
and
A
C
H
I
ACHI
A
C
H
I
. Determine the greatest positive real constant
k
k
k
such that, for any triangle
△
A
B
C
\triangle ABC
△
A
BC
, the following inequality is true:
[
D
E
F
G
H
I
]
≥
k
⋅
[
A
B
C
]
[DEFGHI]\geq k\cdot [ABC]
[
D
EFG
H
I
]
≥
k
⋅
[
A
BC
]
Note:
[
X
]
[X]
[
X
]
denotes the area of polygon
X
X
X
.
2
1
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2017 and a (quadratic) equation
a) Find a pair(s) of integers
(
x
,
y
)
(x,y)
(
x
,
y
)
such that:
y
2
=
x
3
+
2017
y^2=x^3+2017
y
2
=
x
3
+
2017
b) Prove that there isn't integers
x
x
x
and
y
y
y
, with
y
y
y
not divisible by
3
3
3
, such that:
y
2
=
x
3
−
2017
y^2=x^3-2017
y
2
=
x
3
−
2017
1
1
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ATM, 1000 coins and Powers of 2
In certain country, the coins have the following values:
2
0
,
2
1
,
2
2
,
…
2
10
2^0, 2^1, 2^2,\dots 2^{10}
2
0
,
2
1
,
2
2
,
…
2
10
. A cash machine has
1000
1000
1000
coins of each value and give the money using each coin(of each value) at most once. The customers order all the positive integers:
1
,
2
,
3
,
4
,
5
,
…
1,2,3,4,5,\dots
1
,
2
,
3
,
4
,
5
,
…
(in this order) in coins. a) Determine the first integer, such that the cash machine cannot provide. b) In the moment that the first customer can not be attended, by the lack of coins, what are the coins which are not available in the cash machine?