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Contests
National and Regional Contests
PEN Problems
PEN E Problems
23
23
Part of
PEN E Problems
Problems
(1)
E 23
Source:
5/25/2007
Let
p
1
=
2
,
p
2
=
3
,
p
3
=
5
,
⋯
,
p
n
p_{1}=2, p_{2}={3}, p_{3}=5, \cdots, p_{n}
p
1
=
2
,
p
2
=
3
,
p
3
=
5
,
⋯
,
p
n
be the first
n
n
n
prime numbers, where
n
≥
3
n \ge 3
n
≥
3
. Prove that
1
p
1
2
+
1
p
2
2
+
⋯
+
1
p
n
2
+
1
p
1
p
2
⋯
p
n
<
1
2
.
\frac{1}{{p_{1}}^{2}}+\frac{1}{{p_{2}}^{2}}+\cdots+\frac{1}{{p_{n}}^{2}}+\frac{1}{p_{1}p_{2}\cdots p_{n}}< \frac{1}{2}.
p
1
2
1
+
p
2
2
1
+
⋯
+
p
n
2
1
+
p
1
p
2
⋯
p
n
1
<
2
1
.
induction
number theory
prime numbers