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National and Regional Contests
PEN Problems
PEN J Problems
10
J 10
J 10
Source:
May 25, 2007
inequalities
Divisor Functions
Problem Statement
Show that [*] if
n
>
49
n>49
n
>
49
, then there are positive integers
a
>
1
a>1
a
>
1
and
b
>
1
b>1
b
>
1
such that
a
+
b
=
n
a+b=n
a
+
b
=
n
and
ϕ
(
a
)
a
+
ϕ
(
b
)
b
<
1
\frac{\phi(a)}{a}+\frac{\phi(b)}{b}<1
a
ϕ
(
a
)
+
b
ϕ
(
b
)
<
1
. [*] if
n
>
4
n>4
n
>
4
, then there are
a
>
1
a>1
a
>
1
and
b
>
1
b>1
b
>
1
such that
a
+
b
=
n
a+b=n
a
+
b
=
n
and
ϕ
(
a
)
a
+
ϕ
(
b
)
b
>
1
\frac{\phi(a)}{a}+\frac{\phi(b)}{b}>1
a
ϕ
(
a
)
+
b
ϕ
(
b
)
>
1
.
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