MathDB
Problems
Contests
National and Regional Contests
Argentina Contests
Argentina National Olympiad
2001 Argentina National Olympiad
2001 Argentina National Olympiad
Part of
Argentina National Olympiad
Subcontests
(6)
6
1
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cover a rectangle with a finite number of rectangles
Given a rectangle
R
\mathcal{R}
R
of area
100000
100000
100000
, Pancho must completely cover the rectangle
R
\mathcal{R}
R
with a finite number of rectangles with sides parallel to the sides of
R
\mathcal{R}
R
. Next, Martín colors some rectangles of Pancho's cover red so that no two red rectangles have interior points in common. If the red area is greater than
0.00001
0.00001
0.00001
, Martin wins. Otherwise, Pancho wins. Prove that Pancho can cover to ensure victory,
5
1
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set of 50 distinct positive integers
All sets of
49
49
49
distinct positive integers less than or equal to
100
100
100
are considered. Leandro assigned each of these sets a positive integer less than or equal to
100
100
100
. Prove that there is a set
L
L
L
of
50
50
50
distinct positive integers less than or equal to
100
100
100
, such that for each number
x
x
x
of
L
L
L
the number that Leandro assigned to the set of
49
49
49
numbers
L
−
{
x
}
L-\{ x\}
L
−
{
x
}
is different from
x
x
x
. Clarification:
L
−
{
x
}
L-\{x\}
L
−
{
x
}
denotes the set that results from removing the number
x
x
x
from
L
L
L
.
4
1
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k = sum 1/a_i
Find all positive integers
k
k
k
that can be expressed as the sum of
50
50
50
fractions such that the numerators are the
50
50
50
natural numbers from
1
1
1
to
50
50
50
and the denominators are positive integers, that is,
k
=
1
a
1
+
2
a
2
+
…
+
50
a
50
k = \dfrac{1}{a_1} + \dfrac{2}{a_2} + \ldots + \dfrac{50}{a_{50}}
k
=
a
1
1
+
a
2
2
+
…
+
a
50
50
with a
1
,
a
2
,
…
,
a
n
_1 , a_2 , \ldots , a_n
1
,
a
2
,
…
,
a
n
positive integers.
3
1
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five digits 1,4,2,8,6 appear somewhere in decimal expression of a<b
Let
a
a
a
and
b
b
b
be positive integers,
a
<
b
a < b
a
<
b
, such that in the decimal expansion of the fraction
a
b
\dfrac{a}{b}
b
a
the five digits
1
,
4
,
2
,
8
,
6
1,4,2,8,6
1
,
4
,
2
,
8
,
6
appear somewhere, in that order and consecutively. Determine the lowest possible value
b
b
b
can take .
1
1
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7 steps to guess the number
Sergio thinks of a positive integer
S
S
S
, less than or equal to
100
100
100
. Iván must guess the number that Sergio thought of, using the following procedure: in each step, he chooses two positive integers
A
A
A
and
B
B
B
less than
100
100
100
, and asks Sergio what is the greatest common factor between
A
+
S
A+ S
A
+
S
and
B
B
B
. Give a sequence of seven steps that ensures Iván guesses the number
S
S
S
that Sergio thought of.Clarification:In each step, Sergio correctly answers Iván's question.
2
1
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equal angles wanted and given, angle bisector and right angle related
Let
△
A
B
C
\vartriangle ABC
△
A
BC
be a triangle such that angle
∠
A
B
C
\angle ABC
∠
A
BC
is less than angle
∠
A
C
B
\angle ACB
∠
A
CB
. The bisector of angle
∠
B
A
C
\angle BAC
∠
B
A
C
cuts side
B
C
BC
BC
at
D
D
D
. Let
E
E
E
be on side
A
B
AB
A
B
such that
∠
E
D
B
=
9
0
o
\angle EDB = 90^o
∠
E
D
B
=
9
0
o
and
F
F
F
on side
A
C
AC
A
C
such that
∠
B
E
D
=
∠
D
E
F
\angle BED = \angle DEF
∠
BE
D
=
∠
D
EF
. Prove that
∠
B
A
D
=
∠
F
D
C
\angle BAD = \angle FDC
∠
B
A
D
=
∠
F
D
C
.