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National and Regional Contests
Argentina Contests
Argentina National Olympiad
1999 Argentina National Olympiad
1999 Argentina National Olympiad
Part of
Argentina National Olympiad
Subcontests
(6)
6
1
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set E of all fractions 1/n
We consider the set E of all fractions
1
n
\frac{1}{n}
n
1
, where
n
n
n
is a natural number. A maximal arithmetic progression of length
k
k
k
of the set E is an arithmetic progression of
k
k
k
terms such that all its terms belong to the set E, and it is impossible to extend it to the right or to the left with another element of E. For example,
1
20
,
1
8
,
1
5
\frac{1}{20}, \frac{1}{8}, \frac{1}{5}
20
1
,
8
1
,
5
1
, is an arithmetic progression in E of length
3
3
3
, and it is maximal, since to extend it towards to the right you have to add
11
40
\frac{11}{40}
40
11
, which does not belong to E, and to extend it to the left you have to add
−
1
40
\frac{-1}{40}
40
−
1
which does not belong to E either.Prove that for every integer k> 2, there exists a maximal arithmetic progression of length
k
k
k
of the set E.
5
1
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rectangle-shaped puzzle is assembled with 2000 equal rectangle pieces
A rectangle-shaped puzzle is assembled with
2000
2000
2000
pieces that are all equal rectangles, and similar to the large rectangle, so that the sides of the small rectangles are parallel to those of the large one. The shortest side of each piece measures
1
1
1
. Determine what is the minimum possible value of the area of the large rectangle.
4
1
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Coins of diameter 1 have been placed on a square of side 11
Coins of diameter
1
1
1
have been placed on a square of side
11
11
11
, without overlapping or protruding from the square. Can there be
126
126
126
coins? and
127
127
127
? and
128
128
128
?
3
1
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2k people in a trick tournament
In a trick tournament
2
k
2k
2
k
people sign up. All possible matches are played with the condition that in each match, each of the four players knows his partner and does not know any of his two opponents. Determine the maximum number of matches that can be in such a tournament.
1
1
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x c=a+b+186
Three natural numbers greater than or equal to
2
2
2
are written, not necessarily different, and from them a sequence is constructed using the following procedure: in each step, if the penultimate number written is
a
a
a
, the penultimate one is
b
b
b
and the last one is
c
c
c
, it is written
x
x
x
such that
x
⋅
c
=
a
+
b
+
186.
x\cdot c=a+b+186.
x
⋅
c
=
a
+
b
+
186.
Determine all the possible values of the three numbers initially written so that when the process continues indefinitely all the written numbers are natural numbers greater than or equal to
2
2
2
.
2
1
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equal segments, circles, tangents (1999 Argentina OMA L2 p2)
Let
C
1
C_1
C
1
and
C
2
C_2
C
2
be the outer circumferences of centers
O
1
O_1
O
1
and
O
2
O_2
O
2
, respectively. The two tangents to the circumference
C
2
C_2
C
2
are drawn by
O
1
O_1
O
1
, intersecting
C
1
C_1
C
1
at
P
P
P
and
P
′
P'
P
′
. The two tangents to the circumference
C
1
C_1
C
1
are drawn by
O
2
O_2
O
2
, intersecting
C
2
C_2
C
2
at
Q
Q
Q
and
Q
′
Q'
Q
′
. Prove that the segment
P
P
′
PP'
P
P
′
is equal to the segment
Q
Q
′
QQ'
Q
Q
′
.