MathDB
Problems
Contests
National and Regional Contests
PEN Problems
PEN J Problems
PEN J Problems
Part of
PEN Problems
Subcontests
(22)
22
1
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J 22
Let
n
n
n
be an odd positive integer. Prove that
σ
(
n
)
3
<
n
4
\sigma(n)^3 <n^4
σ
(
n
)
3
<
n
4
.
21
1
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J 21
Show that for any positive integer
n
n
n
,
σ
(
n
!
)
n
!
≥
∑
k
=
1
n
1
k
.
\frac{\sigma(n!)}{n!}\ge \sum_{k=1}^{n}\frac{1}{k}.
n
!
σ
(
n
!)
≥
k
=
1
∑
n
k
1
.
20
1
Hide problems
J 20
Show that
σ
(
n
)
−
d
(
m
)
\sigma (n) -d(m)
σ
(
n
)
−
d
(
m
)
is even for all positive integers
m
m
m
and
n
n
n
where
m
m
m
is the largest odd divisor of
n
n
n
.
19
1
Hide problems
J 19
Prove that
σ
(
n
)
ϕ
(
n
)
<
n
2
\sigma(n)\phi(n) < n^2
σ
(
n
)
ϕ
(
n
)
<
n
2
, but that there is a positive constant
c
c
c
such that
σ
(
n
)
ϕ
(
n
)
≥
c
n
2
\sigma(n)\phi(n) \ge c n^2
σ
(
n
)
ϕ
(
n
)
≥
c
n
2
holds for all positive integers
n
n
n
.
18
1
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J 18
Prove that for any
δ
\delta
δ
greater than 1 and any positive number
ϵ
\epsilon
ϵ
, there is an
n
n
n
such that
∣
σ
(
n
)
n
−
δ
∣
<
ϵ
\left \vert \frac{\sigma (n)}{n} -\delta \right \vert < \epsilon
n
σ
(
n
)
−
δ
<
ϵ
.
17
1
Hide problems
J 17
Show that
ϕ
(
n
)
+
σ
(
n
)
≥
2
n
\phi(n)+\sigma(n) \ge 2n
ϕ
(
n
)
+
σ
(
n
)
≥
2
n
for all positive integers
n
n
n
.
16
1
Hide problems
J 16
We say that an integer
m
≥
1
m \ge 1
m
≥
1
is super-abundant if
σ
(
m
)
m
>
σ
(
k
)
k
\frac{\sigma(m)}{m}>\frac{\sigma(k)}{k}
m
σ
(
m
)
>
k
σ
(
k
)
for all
k
∈
{
1
,
2
,
⋯
,
m
−
1
}
k \in \{1, 2,\cdots, m-1 \}
k
∈
{
1
,
2
,
⋯
,
m
−
1
}
. Prove that there exists an infinite number of super-abundant numbers.
15
1
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J 15
Determine all positive integers for which
d
(
n
)
=
n
3
d(n)=\frac{n}{3}
d
(
n
)
=
3
n
holds.
14
1
Hide problems
J 14
Find all positive integers
n
n
n
such that
d
(
n
)
3
=
4
n
{d(n)}^{3} =4n
d
(
n
)
3
=
4
n
.
13
1
Hide problems
J 13
Determine all positive integers
k
k
k
such that
d
(
n
2
)
d
(
n
)
=
k
\frac{d(n^{2})}{d(n)}= k
d
(
n
)
d
(
n
2
)
=
k
for some
n
∈
N
n \in \mathbb{N}
n
∈
N
.
12
1
Hide problems
J 12
Determine all positive integers
n
n
n
such that
n
=
d
(
n
)
2
n={d(n)}^2
n
=
d
(
n
)
2
.
11
1
Hide problems
J 11
Prove that
d
(
(
n
2
+
1
)
2
)
{d((n^2 +1)}^2)
d
((
n
2
+
1
)
2
)
does not become monotonic from any given point onwards.
10
1
Hide problems
J 10
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
.
9
1
Hide problems
J 9
Show that the set of all numbers
ϕ
(
n
+
1
)
ϕ
(
n
)
\frac{\phi(n+1)}{\phi(n)}
ϕ
(
n
)
ϕ
(
n
+
1
)
is dense in the set of all positive reals.
8
1
Hide problems
J 8
Prove that for any
δ
∈
[
0
,
1
]
\delta\in[0,1]
δ
∈
[
0
,
1
]
and any
ε
>
0
\varepsilon>0
ε
>
0
, there is an
n
∈
N
n\in\mathbb{N}
n
∈
N
such that
∣
ϕ
(
n
)
n
−
δ
∣
<
ε
\left |\frac{\phi (n)}{n}-\delta\right| <\varepsilon
n
ϕ
(
n
)
−
δ
<
ε
.
7
1
Hide problems
J 7
Show that if the equation
ϕ
(
x
)
=
n
\phi(x)=n
ϕ
(
x
)
=
n
has one solution, it always has a second solution,
n
n
n
being given and
x
x
x
being the unknown.
6
1
Hide problems
J 6
Show that if
m
m
m
and
n
n
n
are relatively prime positive integers, then
ϕ
(
5
m
−
1
)
≠
5
n
−
1
\phi( 5^m -1) \neq 5^{n}-1
ϕ
(
5
m
−
1
)
=
5
n
−
1
.
5
1
Hide problems
J 5
If
n
n
n
is composite, prove that
ϕ
(
n
)
≤
n
−
n
\phi(n) \le n- \sqrt{n}
ϕ
(
n
)
≤
n
−
n
.
4
1
Hide problems
J 4
Let
m
m
m
,
n
n
n
be positive integers. Prove that, for some positive integer
a
a
a
, each of
ϕ
(
a
)
\phi(a)
ϕ
(
a
)
,
ϕ
(
a
+
1
)
\phi(a+1)
ϕ
(
a
+
1
)
,
⋯
\cdots
⋯
,
ϕ
(
a
+
n
)
\phi(a+n)
ϕ
(
a
+
n
)
is a multiple of
m
m
m
.
3
1
Hide problems
J 3
If
p
p
p
is a prime and
n
n
n
an integer such that
1
<
n
≤
p
1<n \le p
1
<
n
≤
p
, then
ϕ
(
∑
k
=
0
p
−
1
n
k
)
≡
0
(
m
o
d
p
)
.
\phi \left( \sum_{k=0}^{p-1}n^{k}\right) \equiv 0 \; \pmod{p}.
ϕ
(
k
=
0
∑
p
−
1
n
k
)
≡
0
(
mod
p
)
.
2
1
Hide problems
J 2
Show that for all
n
∈
N
n \in \mathbb{N}
n
∈
N
,
n
=
∑
d
∣
n
ϕ
(
d
)
.
n = \sum^{}_{d \vert n}\phi(d).
n
=
d
∣
n
∑
ϕ
(
d
)
.
1
1
Hide problems
J 1
Let
n
n
n
be an integer with
n
≥
2
n \ge 2
n
≥
2
. Show that
ϕ
(
2
n
−
1
)
\phi(2^{n}-1)
ϕ
(
2
n
−
1
)
is divisible by
n
n
n
.