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Problems
Contests
National and Regional Contests
Romania Contests
Romania Team Selection Test
1992 Romania Team Selection Test
2
Romanian TST 1992
Romanian TST 1992
Source: Romanian TST 1992 - Day 2 - Problem 2
April 9, 2012
inequalities
inequalities proposed
Problem Statement
Let
a
1
,
a
2
,
.
.
.
,
a
k
a_1, a_2, ..., a_k
a
1
,
a
2
,
...
,
a
k
be distinct positive integers such that the
2
k
2^k
2
k
sums
∑
i
=
1
k
ϵ
i
a
i
\displaystyle\sum\limits_{i=1}^{k}{\epsilon_i a_i}
i
=
1
∑
k
ϵ
i
a
i
,
ϵ
i
∈
{
0
,
1
}
\epsilon_i\in\left\{0,1\right\}
ϵ
i
∈
{
0
,
1
}
are distinct. a) Show that
1
a
1
+
1
a
2
+
.
.
.
+
1
a
k
≤
2
(
1
−
2
−
k
)
\dfrac{1}{a_1}+\dfrac{1}{a_2}+...+\dfrac{1}{a_k}\le2(1-2^{-k})
a
1
1
+
a
2
1
+
...
+
a
k
1
≤
2
(
1
−
2
−
k
)
; b) Find the sequences
(
a
1
,
a
2
,
.
.
.
,
a
k
)
(a_1,a_2,...,a_k)
(
a
1
,
a
2
,
...
,
a
k
)
for which the equality holds.Șerban Buzețeanu
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