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National and Regional Contests
Portugal Contests
Portugal MO
2011 Portugal MO
2011 Portugal MO
Part of
Portugal MO
Subcontests
(6)
6
1
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n = not sum of 2 or more consecutive natural numbers
The number
1000
1000
1000
can be written as the sum of
16
16
16
consecutive natural numbers:
1000
=
55
+
56
+
.
.
.
+
70.
1000 = 55 + 56 + ... + 70.
1000
=
55
+
56
+
...
+
70.
Determines all natural numbers that cannot be written as the sum of two or more consecutive natural numbers .
4
1
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2 of 14 with same first and last name - 2011 Portugal (OPM) p4
In a class of
14
14
14
boys, each boy was asked how many classmates had the same first name. and how many colleagues had the same last name as them. The numbers
0
,
1
,
2
,
3
,
4
,
5
0, 1, 2, 3, 4, 5
0
,
1
,
2
,
3
,
4
,
5
and
6
6
6
. Proves that there are two colleagues with the same first name and the same last name
3
1
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set of n numbered lights
A set of
n
n
n
lights, numbered
1
1
1
to
n
n
n
, are initially off. At every moment, it is possible to perform one of the following operations:
∙
\bullet
∙
change the state of lamp
1
1
1
,
∙
\bullet
∙
change the state of lamp
2
2
2
, as long as lamp
1
1
1
is on,
∙
\bullet
∙
change the state of lamp
k
>
2
k > 2
k
>
2
, as long as lamp
k
−
1
k - 1
k
−
1
is on and all lamps
1
,
.
.
.
,
k
−
2
1, . . . , k - 2
1
,
...
,
k
−
2
are off. It shows that it is possible, after a certain number of operations, to have only the lamp left on.
1
1
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9digit tel number _ 2011 Portugal (OPM) p1
A nine-digit telephone number abcdefghi is called memorizable if the sequence of four initial digits abcd is repeated in the sequence of the final five digits efghi. How many memorizable numbers of nine digits exist?
5
1
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DE intersects AB,AC at touchpoints of ABC wiht it's incircle
Let
[
A
B
C
]
[ABC]
[
A
BC
]
be a triangle,
D
D
D
be the orthogonal projection of
B
B
B
on the bisector of
∠
A
C
B
\angle ACB
∠
A
CB
and
E
E
E
the orthogonal projection of
C
C
C
on the bisector of
∠
A
B
C
\angle ABC
∠
A
BC
. Prove that
D
E
DE
D
E
intersects the sides
[
A
B
]
[AB]
[
A
B
]
and
[
A
C
]
[AC]
[
A
C
]
at the touchpoints of the circle inscribed in the triangle
[
A
B
C
]
[ABC]
[
A
BC
]
.
2
1
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parallelogram wanted, similar triangles on each side of a triangle
The point
P
P
P
, inside the triangle
[
A
B
C
]
[ABC]
[
A
BC
]
, lies on the perpendicular bisector of
[
A
B
]
[AB]
[
A
B
]
.
Q
Q
Q
and
R
R
R
points, exterior to the triangle, they are such that
[
B
P
A
]
,
[
B
Q
C
]
[BPA], [BQC]
[
BP
A
]
,
[
BQC
]
and
[
C
R
A
]
[CRA]
[
CR
A
]
are similar triangles. Shows that
[
P
Q
C
R
]
[PQCR]
[
PQCR
]
is a parallelogram. https://cdn.artofproblemsolving.com/attachments/f/5/6e036b127f8a013794b8246cbb1544e7280d4a.png