MathDB
Problems
Contests
National and Regional Contests
Argentina Contests
Argentina National Olympiad
2004 Argentina National Olympiad
2004 Argentina National Olympiad
Part of
Argentina National Olympiad
Subcontests
(6)
6
1
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|a_n-n^2 | < n/2, arithmetic progression
Decide if it is possible to generate an infinite sequence of positive integers
a
n
a_n
a
n
such that in the sequence there are no three terms that are in arithmetic progression and that for all
n
n
n
\left |a_n-n^2\right | <\frac{n}{2}. Clarification: Three numbers
a
a
a
,
b
b
b
,
c
c
c
are in arithmetic progression if and only if
2
b
=
a
+
c
2b=a+c
2
b
=
a
+
c
.
4
1
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3 colors in a x b board
Determine all positive integers
a
a
a
and
b
b
b
such that each square on the
a
×
b
a\times b
a
×
b
board can be colored red, blue, or green such that each red square has exactly one blue neighbor and one green neighbor, each blue square has exactly one red and one green neighbor and each green square has exactly one red and one blue neighbor.Clarification: Two squares are neighbors if they have a common side.
3
1
Hide problems
0s and 1s in rectangular board
Zeros and ones are placed in each square of a rectangular board. Such a board is said to be varied if each row contains at least one
0
0
0
and at least two
1
1
1
s. Given n
≥
3
,
\geq 3,
≥
3
,
find all integers
k
>
1
k>1
k
>
1
with the following property:The columns of each varied board of
k
k
k
rows and n columns can be permuted so that in each row of the new board the
1
1
1
s do not form a block (that is, there are at least two
1
1
1
s that are separated by one or more
0
0
0
s).
2
1
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a^2c =b^2d , ab+cd =2^{99}+2^{101}
Determine all positive integers
a
,
b
,
c
,
d
a,b,c,d
a
,
b
,
c
,
d
such that
{
a
<
b
a
2
c
=
b
2
d
a
b
+
c
d
=
2
99
+
2
101
\begin{cases} a<b \\ a^2c =b^2d \\ ab+cd =2^{99}+2^{101} \end{cases}
⎩
⎨
⎧
a
<
b
a
2
c
=
b
2
d
ab
+
c
d
=
2
99
+
2
101
1
1
Hide problems
sequence of 2004 integers [n+k+\sqrt{n+k}]
For each positive integer
n
n
n
we consider the sequence of
2004
2004
2004
integers
[
n
+
n
]
,
[
n
+
1
+
n
+
1
]
,
[
n
+
2
+
n
+
2
]
,
…
,
[
n
+
2003
+
n
+
2003
]
\left [n+\sqrt{n}\right ],\left [n+1+\sqrt{n+1}\right ],\left [n+2+\sqrt{n+2}\right ],\ldots ,\left [n+2003+\sqrt{n+2003}\right ]
[
n
+
n
]
,
[
n
+
1
+
n
+
1
]
,
[
n
+
2
+
n
+
2
]
,
…
,
[
n
+
2003
+
n
+
2003
]
Determine the smallest integer
n
n
n
such that the
2004
2004
2004
numbers in the sequence are
2004
2004
2004
consecutive integers. Clarification: The brackets indicate the integer part.
5
1
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area of pentagon ABCDE, AB=BC, CD=DE, <ABC=120^o, <CDE=60^o, BD=2
The pentagon
A
B
C
D
E
ABCDE
A
BC
D
E
has
A
B
=
B
C
,
C
D
=
D
E
,
∠
A
B
C
=
12
0
o
,
∠
C
D
E
=
6
0
o
AB = BC, CD = DE, \angle ABC = 120^o, \angle CDE = 60^o
A
B
=
BC
,
C
D
=
D
E
,
∠
A
BC
=
12
0
o
,
∠
C
D
E
=
6
0
o
and
B
D
=
2
BD = 2
B
D
=
2
. Calculate the area of the pentagon.