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Problems
Contests
National and Regional Contests
Argentina Contests
Argentina National Olympiad
1994 Argentina National Olympiad
1994 Argentina National Olympiad
Part of
Argentina National Olympiad
Subcontests
(6)
6
1
Hide problems
special rectangles in 9x9 board
A
9
×
9
9\times 9
9
×
9
board has a number written on each square: all squares in the first row have
1
1
1
, all squares in the second row have
2
2
2
,
…
\ldots
…
, all squares in the ninth row have
9
9
9
. We will call special rectangle any rectangle of
2
×
3
2\times 3
2
×
3
or
3
×
2
3\times 2
3
×
2
or
4
×
5
4\times 5
4
×
5
or
5
×
4
5\times 4
5
×
4
on the board. The permitted operations are:
∙
\bullet
∙
Simultaneously add
1
1
1
to all the numbers located in a special rectangle.
∙
\bullet
∙
Simultaneously subtract
1
1
1
from all numbers located in a special rectangle.Demonstrate that it is possible to achieve, through a succession of permitted operations, that
80
80
80
squares to have
0
0
0
(zero). What number is left in the remaining box?
5
1
Hide problems
infinite set of points
Let
A
A
A
be an infinite set of points in the plane such that inside each circle there are only a finite number of points of
A
A
A
, with the following properties:
∙
\bullet
∙
(
0
,
0
)
(0, 0)
(
0
,
0
)
belongs to
A
A
A
.
∙
\bullet
∙
If
(
a
,
b
)
(a, b)
(
a
,
b
)
and
(
c
,
d
)
(c, d)
(
c
,
d
)
belong to
A
A
A
, then
(
a
−
c
,
b
−
d
)
(a-c, b-d)
(
a
−
c
,
b
−
d
)
belongs to
A
A
A
.
∙
\bullet
∙
There is a value of
α
\alpha
α
such that by rotating the set
A
A
A
with center at
(
0
,
0
)
(0, 0)
(
0
,
0
)
and angle
α
\alpha
α
, the set
A
A
A
is obtained again. Prove that
α
\alpha
α
must be equal to
±
6
0
∘
\pm 60^{\circ}
±
6
0
∘
or
±
9
0
∘
\pm 90^{\circ}
±
9
0
∘
or
±
12
0
∘
\pm 120^{\circ}
±
12
0
∘
or
±
18
0
∘
\pm 180^{\circ}
±
18
0
∘
.
4
1
Hide problems
rectangle is divided into 9 small rectangles
A rectangle is divided into
9
9
9
small rectangles if by parallel lines to its sides, as shown in the figure. https://cdn.artofproblemsolving.com/attachments/e/d/1fd545862a3c7950249ec54a631c74e59fb9ed.png The four numbers written indicate the areas of the four corresponding rectangles. Prove that the total area of the rectangle is greater than or equal to
90
90
90
.
3
1
Hide problems
two squares, one given, one wanted
Given in the plane the square
A
B
C
D
ABCD
A
BC
D
, the square
A
1
B
1
C
1
D
1
A_1B_1C_1D_1
A
1
B
1
C
1
D
1
, smaller than the first, and a quadrilateral
P
Q
R
S
PQRS
PQRS
that satisfy the following conditions
∙
\bullet
∙
A
B
C
D
ABCD
A
BC
D
and
A
1
B
1
C
1
D
1
A_1B_1C_1D_1
A
1
B
1
C
1
D
1
have a common center and respectively parallel sides.
∙
\bullet
∙
P
P
P
,
Q
Q
Q
,
R
R
R
,
S
S
S
belong one to each side of the square
A
B
C
D
ABCD
A
BC
D
.
∙
\bullet
∙
A
1
A_1
A
1
,
B
1
B_1
B
1
,
C
1
C_1
C
1
,
D
1
D_1
D
1
belong one to each side of the quadrilateral
P
Q
R
S
PQRS
PQRS
. Prove that
P
Q
R
S
PQRS
PQRS
is a square.
2
1
Hide problems
x^4 + 6x^3 + 11x^2 + 3x + 31 is a perfect square
For what positive integer values of
x
x
x
is
x
4
+
6
x
3
+
11
x
2
+
3
x
+
31
x^4 + 6x^3 + 11x^2 + 3x + 31
x
4
+
6
x
3
+
11
x
2
+
3
x
+
31
a perfect square?
1
1
Hide problems
30 segments of lengths 1, \sqrt3, \sqrt5, \sqrt7, ..., \sqrt{59}
30
30
30
segments of lengths1, \sqrt{3}, \sqrt{5}, \sqrt{7}, \sqrt{9}, \ldots , \sqrt{59} have been drawn on a blackboard. In each step, two of the segments are deleted and a new segment of length equal to the hypotenuse of the right triangle with legs equal to the two deleted segments is drawn. After
29
29
29
steps only one segment remains. Find the possible values of its length.