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National and Regional Contests
China Contests
China Team Selection Test
2018 China Team Selection Test
5
Sad Inequality
Sad Inequality
Source: 2018 China TST Day 4 Q2
January 21, 2018
inequalities
Problem Statement
Given positive integers
n
,
k
n, k
n
,
k
such that
n
≥
4
k
n\ge 4k
n
≥
4
k
, find the minimal value
λ
=
λ
(
n
,
k
)
\lambda=\lambda(n,k)
λ
=
λ
(
n
,
k
)
such that for any positive reals
a
1
,
a
2
,
…
,
a
n
a_1,a_2,\ldots,a_n
a
1
,
a
2
,
…
,
a
n
, we have
∑
i
=
1
n
a
i
a
i
2
+
a
i
+
1
2
+
⋯
+
a
i
+
k
2
≤
λ
\sum\limits_{i=1}^{n} {\frac{{a}_{i}}{\sqrt{{a}_{i}^{2}+{a}_{{i}+{1}}^{2}+{\cdots}{{+}}{a}_{{i}{+}{k}}^{2}}}} \le \lambda
i
=
1
∑
n
a
i
2
+
a
i
+
1
2
+
⋯
+
a
i
+
k
2
a
i
≤
λ
Where
a
n
+
i
=
a
i
,
i
=
1
,
2
,
…
,
k
a_{n+i}=a_i,i=1,2,\ldots,k
a
n
+
i
=
a
i
,
i
=
1
,
2
,
…
,
k
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