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Contests
National and Regional Contests
Portugal Contests
Portugal MO
2005 Portugal MO
2005 Portugal MO
Part of
Portugal MO
Subcontests
(6)
6
1
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f(a + b)f(a - b) = f(a^2) in N
Prove that there is a unique function
f
:
N
→
N
f: N\to N
f
:
N
→
N
, that verifies
f
(
a
+
b
)
f
(
a
−
b
)
=
f
(
a
2
)
f(a + b)f(a - b) = f(a^2)
f
(
a
+
b
)
f
(
a
−
b
)
=
f
(
a
2
)
, for any
a
,
b
∈
N
a, b\in N
a
,
b
∈
N
such that
a
>
b
a > b
a
>
b
.
4
1
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even number k>46 is sum of 2 numbers whose sum of divisors is > 2n
A natural number
n
n
n
is said to be abundant if the sum of its divisors is greater than
2
n
2n
2
n
. For example,
18
18
18
is abundant because the sum of its divisors,
1
+
2
+
3
+
6
+
9
+
18
1 + 2 + 3 + 6 + 9 + 18
1
+
2
+
3
+
6
+
9
+
18
, is greater than
36
36
36
. Write every even number greater than
46
46
46
as a sum of two numbers abundant.
3
1
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a switch and a light bulb at each square of xa x b board
On a board with
a
a
a
rows and
b
b
b
columns, each square has a switch and an unlit light bulb. By pressing the switch of a house, the lamp in that house changes state, along with the lamps in the same row and those in the same column (those that are on go out and the that are off light up). For what values of
a
a
a
and
b
b
b
is it possible to have just one lamp on, by pressing a series of switches?
1
1
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2005 people at a SuperRockPop concert
In line for a SuperRockPop concert were 2005 people. With the aim of offering
3
3
3
tickets for the "backstage", the first person in line was asked to shout "Super", ` the second "Rock", ` the third "Pop", ` the fourth "Super", ` the fifth "Rock", ` the sixth "Pop" and so on. Anyone who said "Rock" or "Pop" was eliminated. This process was repeated, always starting from the first person in the new line, until only
3
3
3
people remained. What positions were these people in at the beginning?
5
1
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AE x FC = BF x ED for bicentric ABCD 2005 Portugal p5
Considers a quadrilateral
[
A
B
C
D
]
[ABCD]
[
A
BC
D
]
that has an inscribed circle and a circumscribed circle. The sides
[
A
D
]
[AD]
[
A
D
]
and
[
B
C
]
[BC]
[
BC
]
are tangent to the circle inscribed at points
E
E
E
and
F
F
F
, respectively. Prove that
A
E
⋅
F
C
=
B
F
⋅
E
D
AE \cdot F C = BF \cdot ED
A
E
⋅
FC
=
BF
⋅
E
D
. https://1.bp.blogspot.com/-6o1fFTdZ69E/X4XMo98ndAI/AAAAAAAAMno/7FXiJnWzJgcfSn-qSRoEAFyE8VgxmeBjwCLcBGAsYHQ/s0/2005%2BPortugal%2Bp5.png
2
1
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AB <BD wanted, two right triangles, midpoint 2004 Portugal p2
Consider the triangles
[
A
B
C
]
[ABC]
[
A
BC
]
and
[
E
D
C
]
[EDC]
[
E
D
C
]
, right at
A
A
A
and
D
D
D
, respectively. Show that, if
E
E
E
is the midpoint of
[
A
C
]
[AC]
[
A
C
]
, then
A
B
<
B
D
AB <BD
A
B
<
B
D
. https://cdn.artofproblemsolving.com/attachments/c/3/75bc1bda1a22bcf00d4fe7680c80a81a9aaa4c.png