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Contests
National and Regional Contests
Malaysia Contests
JOM Shortlists
JOM 2015 Shortlist
A9
A9
Part of
JOM 2015 Shortlist
Problems
(1)
Inequality + Sequence
Source: Junior Olympiad of Malaysia Shortlist 2015 A9
7/17/2015
Let
2
n
2n
2
n
positive reals
a
1
,
a
2
,
⋯
,
a
n
,
b
1
,
b
2
,
⋯
,
b
n
a_1, a_2, \cdots, a_n, b_1, b_2, \cdots, b_n
a
1
,
a
2
,
⋯
,
a
n
,
b
1
,
b
2
,
⋯
,
b
n
satisfy
a
i
+
1
≥
2
a
i
a_{i+1}\ge 2a_i
a
i
+
1
≥
2
a
i
and
b
i
+
1
≤
b
i
b_{i+1} \le b_i
b
i
+
1
≤
b
i
for
1
≤
i
≤
n
−
1
1\le i\le n-1
1
≤
i
≤
n
−
1
. Find the least constant
C
C
C
that satisfy:
∑
i
=
1
n
a
i
b
i
≥
C
(
a
1
+
a
2
+
⋯
+
a
n
)
b
1
+
b
2
+
⋯
+
b
n
\displaystyle \sum^{n}_{i=1}{\frac{a_i}{b_i}} \ge \displaystyle \frac{C(a_1+a_2+\cdots+a_n)}{b_1+b_2+\cdots+b_n}
i
=
1
∑
n
b
i
a
i
≥
b
1
+
b
2
+
⋯
+
b
n
C
(
a
1
+
a
2
+
⋯
+
a
n
)
and determine all equality case with that constant
C
C
C
.
inequalities