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International Contests
APMO
1997 APMO
1
Sum
Sum
Source: APMO 1997
March 17, 2006
inequalities
Problem Statement
Given:
S
=
1
+
1
1
+
1
3
+
1
1
+
1
3
+
1
6
+
⋯
+
1
1
+
1
3
+
1
6
+
⋯
+
1
1993006
S = 1 + \frac{1}{1 + \frac{1}{3}} + \frac{1}{1 + \frac{1}{3} + \frac{1} {6}} + \cdots + \frac{1}{1 + \frac{1}{3} + \frac{1}{6} + \cdots + \frac{1} {1993006}}
S
=
1
+
1
+
3
1
1
+
1
+
3
1
+
6
1
1
+
⋯
+
1
+
3
1
+
6
1
+
⋯
+
1993006
1
1
where the denominators contain partial sums of the sequence of reciprocals of triangular numbers (i.e.
k
=
n
(
n
+
1
)
2
k=\frac{n(n+1)}{2}
k
=
2
n
(
n
+
1
)
for
n
=
1
n = 1
n
=
1
,
2
2
2
,
…
\ldots
…
,
1996
1996
1996
). Prove that
S
>
1001
S>1001
S
>
1001
.
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