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International Contests
IMO Longlists
1982 IMO Longlists
28
28
Part of
1982 IMO Longlists
Problems
(1)
The inequality for (u1,u2,...,un) n-tuple
Source: IMO LongList 1982 - P28
3/18/2011
Let
(
u
1
,
…
,
u
n
)
(u_1, \ldots, u_n)
(
u
1
,
…
,
u
n
)
be an ordered
n
n
n
tuple. For each
k
,
1
≤
k
≤
n
k, 1 \leq k \leq n
k
,
1
≤
k
≤
n
, define
v
k
=
u
1
u
2
⋯
u
k
k
v_k=\sqrt[k]{u_1u_2 \cdots u_k}
v
k
=
k
u
1
u
2
⋯
u
k
. Prove that
∑
k
=
1
n
v
k
≤
e
⋅
∑
k
=
1
n
u
k
.
\sum_{k=1}^n v_k \leq e \cdot \sum_{k=1}^n u_k.
k
=
1
∑
n
v
k
≤
e
⋅
k
=
1
∑
n
u
k
.
(
e
e
e
is the base of the natural logarithm).
inequalities
inequalities unsolved