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Contests
National and Regional Contests
Argentina Contests
Argentina National Olympiad
2008 Argentina National Olympiad
2008 Argentina National Olympiad
Part of
Argentina National Olympiad
Subcontests
(6)
6
1
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black, white and green squares, starting with BW squares in a x b board
Consider a board of
a
×
b
a \times b
a
×
b
, with
a
a
a
and
b
b
b
integers greater than or equal to
2
2
2
. Initially their squares are colored black and white like a chess board. The permitted operation consists of choosing two squares with a common side and recoloring them as follows: a white square becomes black; a black box turns green; a green box turns white. Determine for which values of
a
a
a
and
b
b
b
it is possible, by a succession of allowed operations, to make all the squares that were initially white end black and all the squares that were initially black end white. Clarification: Initially there are no green squares, but they appear after the first operation.
3
1
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area wanted, circle of center O, <AOB = 120^o,AD = 2, BD = 1, CD = \sqrt2
On a circle of center
O
O
O
, let
A
A
A
and
B
B
B
be points on the circle such that
∠
A
O
B
=
12
0
o
\angle AOB = 120^o
∠
A
OB
=
12
0
o
. Point
C
C
C
lies on the small arc
A
B
AB
A
B
and point
D
D
D
lies on the segment
A
B
AB
A
B
. Let also
A
D
=
2
,
B
D
=
1
AD = 2, BD = 1
A
D
=
2
,
B
D
=
1
and
C
D
=
2
CD = \sqrt2
C
D
=
2
. Calculate the area of triangle
A
B
C
ABC
A
BC
.
1
1
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101 numbers in a line
101
101
101
positive integers are written on a line. Prove that we can write signs \plus{}, signs
×
\times
×
and parenthesis between them, without changing the order of the numbers, in such a way that the resulting expression makes sense and the result is divisible by
16
!
16!
16
!
.
2
1
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Numbers on a board
In every cell of a
60
×
60
60 \times 60
60
×
60
board is written a real number, whose absolute value is less or equal than
1
1
1
. The sum of all numbers on the board equals
600
600
600
. Prove that there is a
12
×
12
12 \times 12
12
×
12
square in the board such that the absolute value of the sum of all numbers on it is less or equal than
24
24
24
.
4
1
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[2x]+[3x]+[7x]=2008
Find all real numbers
x
x
x
which satisfy the following equation: [2x]\plus{}[3x]\plus{}[7x]\equal{}2008. Note:
[
x
]
[x]
[
x
]
means the greatest integer less or equal than
x
x
x
.
5
1
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Perfect powers ending in 2008
Find all perfect powers whose last
4
4
4
digits are
2
,
0
,
0
,
8
2,0,0,8
2
,
0
,
0
,
8
, in that order.