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China Contests
XES Mathematics Olympiad
the 9th XMO
1
1
Part of
the 9th XMO
Problems
(1)
9th XMO 2022 P1: On bounding the product of Sum and Harmonic
Source: 9th XMO 2022
6/18/2022
For any
n
n
n
consecutive integers
a
1
,
⋯
,
a
n
a_1, \cdots, a_n
a
1
,
⋯
,
a
n
, prove that
(
a
1
+
⋯
+
a
n
)
⋅
(
1
a
1
+
⋯
+
1
a
n
)
⩽
n
(
n
+
1
)
ln
(
e
n
)
2
.
(a_1+\cdots+a_n)\cdot\left(\frac{1}{a_1}+\cdots+\frac{1}{a_n}\right)\leqslant \frac{n(n+1)\ln(\text{e}n)}{2}.
(
a
1
+
⋯
+
a
n
)
⋅
(
a
1
1
+
⋯
+
a
n
1
)
⩽
2
n
(
n
+
1
)
ln
(
e
n
)
.
algebra
inequalities
China