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Taiwan National Olympiad
1997 Taiwan National Olympiad
3
3
Part of
1997 Taiwan National Olympiad
Problems
(1)
n of reals
Source: 6-th Taiwanese Mathematical Olympiad 1997
1/18/2007
Let
n
>
2
n>2
n
>
2
be an integer. Suppose that
a
1
,
a
2
,
.
.
.
,
a
n
a_{1},a_{2},...,a_{n}
a
1
,
a
2
,
...
,
a
n
are real numbers such that
k
i
=
a
i
−
1
+
a
i
+
1
a
i
k_{i}=\frac{a_{i-1}+a_{i+1}}{a_{i}}
k
i
=
a
i
a
i
−
1
+
a
i
+
1
is a positive integer for all
i
i
i
(Here
a
0
=
a
n
,
a
n
+
1
=
a
1
a_{0}=a_{n},a_{n+1}=a_{1}
a
0
=
a
n
,
a
n
+
1
=
a
1
). Prove that
2
n
≤
a
1
+
a
2
+
.
.
.
+
a
n
≤
3
n
2n\leq a_{1}+a_{2}+...+a_{n}\leq 3n
2
n
≤
a
1
+
a
2
+
...
+
a
n
≤
3
n
.
induction
inequalities proposed
inequalities