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National and Regional Contests
Korea Contests
Korea National Olympiad
2005 Korea National Olympiad
7
7
Part of
2005 Korea National Olympiad
Problems
(1)
19th kmo #7
Source: KMO round 2, problem 7
2/3/2006
For a positive integer
n
n
n
, let
f
(
n
)
f(n)
f
(
n
)
be the number of factors of
n
2
+
n
+
1
n^2+n+1
n
2
+
n
+
1
. Show that there are infinitely many integers
n
n
n
which satisfy
f
(
n
)
≥
f
(
n
+
1
)
f(n) \geq f(n+1)
f
(
n
)
≥
f
(
n
+
1
)
.
number theory unsolved
number theory