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Contests
National and Regional Contests
Canada Contests
Canada National Olympiad
2014 Canada National Olympiad
1
Sequence a with product 1
Sequence a with product 1
Source: 2014 CMO #1
May 11, 2014
induction
inequalities
inequalities proposed
Problem Statement
Let
a
1
,
a
2
,
…
,
a
n
a_1,a_2,\dots,a_n
a
1
,
a
2
,
…
,
a
n
be positive real numbers whose product is
1
1
1
. Show that the sum
a
1
1
+
a
1
+
a
2
(
1
+
a
1
)
(
1
+
a
2
)
+
a
3
(
1
+
a
1
)
(
1
+
a
2
)
(
1
+
a
3
)
+
⋯
+
a
n
(
1
+
a
1
)
(
1
+
a
2
)
⋯
(
1
+
a
n
)
\textstyle\frac{a_1}{1+a_1}+\frac{a_2}{(1+a_1)(1+a_2)}+\frac{a_3}{(1+a_1)(1+a_2)(1+a_3)}+\cdots+\frac{a_n}{(1+a_1)(1+a_2)\cdots(1+a_n)}
1
+
a
1
a
1
+
(
1
+
a
1
)
(
1
+
a
2
)
a
2
+
(
1
+
a
1
)
(
1
+
a
2
)
(
1
+
a
3
)
a
3
+
⋯
+
(
1
+
a
1
)
(
1
+
a
2
)
⋯
(
1
+
a
n
)
a
n
is greater than or equal to
2
n
−
1
2
n
\frac{2^n-1}{2^n}
2
n
2
n
−
1
.
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