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National and Regional Contests
Mongolia Contests
Mongolian Mathematical Olympiad
2001 Mongolian Mathematical Olympiad
Problem 2
inequality in two sequences
inequality in two sequences
Source: Mongolia MO 2001 Teachers P2
April 12, 2021
inequalities
Sequence
algebra
Problem Statement
For positive real numbers
b
1
,
b
2
,
…
,
b
n
b_1,b_2,\ldots,b_n
b
1
,
b
2
,
…
,
b
n
define
a
1
=
b
1
b
1
+
b
2
+
…
+
b
n
and
a
k
=
b
1
+
…
+
b
k
b
1
+
…
+
b
k
−
1
for
k
>
1.
a_1=\frac{b_1}{b_1+b_2+\ldots+b_n}\enspace\text{ and }\enspace a_k=\frac{b_1+\ldots+b_k}{b_1+\ldots+b_{k-1}}\text{ for }k>1.
a
1
=
b
1
+
b
2
+
…
+
b
n
b
1
and
a
k
=
b
1
+
…
+
b
k
−
1
b
1
+
…
+
b
k
for
k
>
1.
Prove that
a
1
+
a
2
+
…
+
a
n
≤
1
a
1
+
1
a
2
+
…
+
1
a
n
a_1+a_2+\ldots+a_n\le\frac1{a_1}+\frac1{a_2}+\ldots+\frac1{a_n}
a
1
+
a
2
+
…
+
a
n
≤
a
1
1
+
a
2
1
+
…
+
a
n
1
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