MathDB
Problems
Contests
National and Regional Contests
USA Contests
USA - College-Hosted Events
Harvard-MIT Mathematics Tournament
2022 Harvard-MIT Mathematics Tournament
4
2022 Team 4
2022 Team 4
Source:
March 14, 2022
algebra
inequalities
Problem Statement
Suppose
n
≥
3
n \ge 3
n
≥
3
is a positive integer. Let
a
1
<
a
2
<
.
.
.
<
a
n
a_1 < a_2 < ... < a_n
a
1
<
a
2
<
...
<
a
n
be an increasing sequence of positive real numbers, and let
a
n
+
1
=
a
1
a_{n+1} = a_1
a
n
+
1
=
a
1
. Prove that
∑
k
=
1
n
a
k
a
k
+
1
>
∑
k
=
1
n
a
k
+
1
a
k
\sum_{k=1}^{n}\frac{a_k}{a_{k+1}}>\sum_{k=1}^{n}\frac{a_{k+1}}{a_k}
k
=
1
∑
n
a
k
+
1
a
k
>
k
=
1
∑
n
a
k
a
k
+
1
Back to Problems
View on AoPS