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National and Regional Contests
Argentina Contests
Argentina National Olympiad
2004 Argentina National Olympiad
1
1
Part of
2004 Argentina National Olympiad
Problems
(1)
sequence of 2004 integers [n+k+\sqrt{n+k}]
Source: Argentina 2004 OMA L3 p1
5/12/2024
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.
number theory
algebra