MathDB
Problems
Contests
National and Regional Contests
Argentina Contests
Argentina National Olympiad
2003 Argentina National Olympiad
2003 Argentina National Olympiad
Part of
Argentina National Olympiad
Subcontests
(6)
6
1
Hide problems
divide set of all positive divisors of 30^n into groups of 3
Determine the positive integers
n
n
n
such that the set of all positive divisors of
3
0
n
30^n
3
0
n
can be divided into groups of three so that the product of the three numbers in each group is the same.
5
1
Hide problems
+ - sings in front of each of numbers 1,2,...,50
Carlos and Yue play the following game: First Carlos writes a
+
+
+
sign or a
−
-
−
sign in front of each of the
50
50
50
numbers
1
,
2
,
⋯
,
50
1,2,\cdots,50
1
,
2
,
⋯
,
50
. Then, in turns, each one chooses a number from the sequence obtained; Start by choosing Yue. If the absolute value of the sum of the
25
25
25
numbers that Carlos chose is greater than or equal to the absolute value of the sum of the
25
25
25
numbers that Yue chose, Carlos wins. In the other case, Yue wins. Determine which of the two players can develop a strategy that will ensure victory, no matter how well their opponent plays, and describe said strategy.
3
1
Hide problems
a=\frac{x_1^2+x_2^2+... + x_n^ 2}{x_1x_2 ... x_n}
Let
a
≥
4
a\geq 4
a
≥
4
be a positive integer. Determine the smallest value of
n
≥
5
n\geq 5
n
≥
5
,
n
≠
a
n\neq a
n
=
a
, such that
a
a
a
can be represented in the form
a
=
x
1
2
+
x
2
2
+
⋯
+
x
n
2
x
1
x
2
⋯
x
n
a=\frac{x_1^2+x_2^2+\cdots + x_n^ 2}{x_1x_2\cdots x_n}
a
=
x
1
x
2
⋯
x
n
x
1
2
+
x
2
2
+
⋯
+
x
n
2
for a suitable choice of the
n
n
n
positive integers
x
1
,
x
2
,
…
,
x
n
x_1,x_2,\ldots ,x_n
x
1
,
x
2
,
…
,
x
n
.
2
1
Hide problems
2003 integers from 1 to 2003 on blackboard
On the blackboard are written the
2003
2003
2003
integers from
1
1
1
to
2003
2003
2003
. Lucas must delete
90
90
90
numbers. Next, Mauro must choose
37
37
37
from the numbers that remain written. If the
37
37
37
numbers Mauro chooses form an arithmetic progression, Mauro wins. If not, Lucas wins. Decide if Lucas can choose the
90
90
90
numbers he erases so that victory is assured.
1
1
Hide problems
1/ [x] - 1/[2x]= 1/6{x}
Find all positive numbers
x
x
x
such that:
1
[
x
]
−
1
[
2
x
]
=
1
6
{
x
}
\frac{1}{[x]}-\frac{1}{[2x]}=\frac{1}{6\{x\}}
[
x
]
1
−
[
2
x
]
1
=
6
{
x
}
1
where
[
x
]
[x]
[
x
]
represents the integer part of
x
x
x
and
{
x
}
=
x
−
[
x
]
\{x\}=x-[x]
{
x
}
=
x
−
[
x
]
.
4
1
Hide problems
area of triangle of circumcenters, right trapezoid related
The trapezoid
A
B
C
D
ABCD
A
BC
D
of bases
A
B
AB
A
B
and
C
D
CD
C
D
, has
∠
A
=
9
0
o
,
A
B
=
6
,
C
D
=
3
\angle A = 90^o, AB = 6, CD = 3
∠
A
=
9
0
o
,
A
B
=
6
,
C
D
=
3
and
A
D
=
4
AD = 4
A
D
=
4
. Let
E
,
G
,
H
E, G, H
E
,
G
,
H
be the circumcenters of triangles
A
B
C
,
A
C
D
,
A
B
D
ABC, ACD, ABD
A
BC
,
A
C
D
,
A
B
D
, respectively. Find the area of the triangle
E
G
H
EGH
EG
H
.