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National and Regional Contests
Portugal Contests
Portugal MO
1999 Portugal MO
1999 Portugal MO
Part of
Portugal MO
Subcontests
(6)
5
1
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4 |n if a_1a_2 + a_2a_3 + ··· + a_{n-1}a_n + a_na_1 = 0, for a_i \in {1,-1}
Each of the numbers
a
1
,
.
.
.
,
a
n
a_1,...,a_n
a
1
,
...
,
a
n
is equal to
1
1
1
or
−
1
-1
−
1
. If
a
1
a
2
+
a
2
a
3
+
⋅
⋅
⋅
+
a
n
−
1
a
n
+
a
n
a
1
=
0
a_1a_2 + a_2a_3 + ··· + a_{n-1}a_n + a_na_1 = 0
a
1
a
2
+
a
2
a
3
+
⋅⋅⋅
+
a
n
−
1
a
n
+
a
n
a
1
=
0
, proves that
n
n
n
is divisible by
4
4
4
.
4
1
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n -> (sum of digits (n^2) ) + 1, repeat
Given a number, we calculate its square and add
1
1
1
to the sum of the digits in this square, obtaining a new number. If we start with the number
7
7
7
we will obtain, in the first step, the number
1
+
(
4
+
9
)
=
14
1+(4+9)=14
1
+
(
4
+
9
)
=
14
, since
7
2
=
49
7^2 = 49
7
2
=
49
. What number will we obtain in the
1999
1999
1999
th step?
2
1
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(2n^2+4n+18)/(3n+3) is integer - Portugal OPM 1999 p2
How many positive integers are there such that
2
n
2
+
4
n
+
18
3
n
+
3
\frac{2n^2+4n+18}{3n+3}
3
n
+
3
2
n
2
+
4
n
+
18
is an integer?
1
1
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balanced 3-digit numbers - Portugal OPM 1999 p1
A number is said to be balanced if one of its digits is average of the others. How many balanced
3
3
3
-digit numbers are there?
6
1
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angle wanted, trisection point, equal angles 1999 Portugal p6
In the triangle
[
A
B
C
]
,
D
[ABC], D
[
A
BC
]
,
D
is the midpoint of
[
A
B
]
[AB]
[
A
B
]
and
E
E
E
is the trisection point of
[
B
C
]
[BC]
[
BC
]
closer to
C
C
C
. If
∠
A
D
C
=
∠
B
A
E
\angle ADC= \angle BAE
∠
A
D
C
=
∠
B
A
E
, find the measue of
∠
B
A
C
\angle BAC
∠
B
A
C
.
3
1
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computational with parallel chords 1999 Portugal p3
If two parallel chords of a circumference,
10
10
10
mm and
14
14
14
mm long, with distance
6
6
6
mm from each other, how long is the chord equidistant from these two?