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Problems
Contests
National and Regional Contests
PEN Problems
PEN P Problems
PEN P Problems
Part of
PEN Problems
Subcontests
(43)
43
1
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P 43
A positive integer
n
n
n
is abundant if the sum of its proper divisors exceeds
n
n
n
. Show that every integer greater than
89
×
315
89 \times 315
89
×
315
is the sum of two abundant numbers.
42
1
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P 42
Prove that for each positive integer
K
K
K
there exist infinitely many even positive integers which can be written in more than
K
K
K
ways as the sum of two odd primes.
41
1
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P 41
The famous conjecture of Goldbach is the assertion that every even integer greater than
2
2
2
is the sum of two primes. Except
2
2
2
,
4
4
4
, and
6
6
6
, every even integer is a sum of two positive composite integers:
n
=
4
+
(
n
−
4
)
n=4+(n-4)
n
=
4
+
(
n
−
4
)
. What is the largest positive even integer that is not a sum of two odd composite integers?
40
1
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P 40
Show that [*] infinitely many perfect squares are a sum of a perfect square and a prime number, [*] infinitely many perfect squares are not a sum of a perfect square and a prime number.
39
1
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P 39
In how many ways can
2
n
2^{n}
2
n
be expressed as the sum of four squares of natural numbers?
38
1
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P 38
Find the smallest possible
n
n
n
for which there exist integers
x
1
x_{1}
x
1
,
x
2
x_{2}
x
2
,
⋯
\cdots
⋯
,
x
n
x_{n}
x
n
such that each integer between
1000
1000
1000
and
2000
2000
2000
(inclusive) can be written as the sum (without repetition), of one or more of the integers
x
1
x_{1}
x
1
,
x
2
x_{2}
x
2
,
⋯
\cdots
⋯
,
x
n
x_{n}
x
n
.
37
1
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P 37
Let
S
n
=
{
1
,
n
,
n
2
,
n
3
,
⋯
}
S_{n}=\{1,n,n^{2},n^{3}, \cdots \}
S
n
=
{
1
,
n
,
n
2
,
n
3
,
⋯
}
, where
n
n
n
is an integer greater than
1
1
1
. Find the smallest number
k
=
k
(
n
)
k=k(n)
k
=
k
(
n
)
such that there is a number which may be expressed as a sum of
k
k
k
(possibly repeated) elements in
S
n
S_{n}
S
n
in more than one way. (Rearrangements are considered the same.)
36
1
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P 36
Let
k
k
k
and
s
s
s
be odd positive integers such that
3
k
−
2
−
1
≤
s
≤
4
k
.
\sqrt{3k-2}-1 \le s \le \sqrt{4k}.
3
k
−
2
−
1
≤
s
≤
4
k
.
Show that there are nonnegative integers
t
t
t
,
u
u
u
,
v
v
v
, and
w
w
w
such that
k
=
t
2
+
u
2
+
v
2
+
w
2
,
and
s
=
t
+
u
+
v
+
w
.
k=t^{2}+u^{2}+v^{2}+w^{2}, \;\; \text{and}\;\; s=t+u+v+w.
k
=
t
2
+
u
2
+
v
2
+
w
2
,
and
s
=
t
+
u
+
v
+
w
.
35
1
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P 35
Prove that every positive integer which is not a member of the infinite set below is equal to the sum of two or more distinct numbers of the set
{
3
,
−
2
,
2
2
3
,
−
2
3
,
⋯
,
2
2
k
3
,
−
2
2
k
+
1
,
⋯
}
=
{
3
,
−
2
,
12
,
−
8
,
48
,
−
32
,
192
,
⋯
}
.
\{ 3,-2, 2^{2}3,-2^{3}, \cdots, 2^{2k}3,-2^{2k+1}, \cdots \}=\{3,-2, 12,-8, 48,-32, 192, \cdots \}.
{
3
,
−
2
,
2
2
3
,
−
2
3
,
⋯
,
2
2
k
3
,
−
2
2
k
+
1
,
⋯
}
=
{
3
,
−
2
,
12
,
−
8
,
48
,
−
32
,
192
,
⋯
}
.
34
1
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P 34
If
n
n
n
is a positive integer which can be expressed in the form
n
=
a
2
+
b
2
+
c
2
n=a^{2}+b^{2}+c^{2}
n
=
a
2
+
b
2
+
c
2
, where
a
,
b
,
c
a, b, c
a
,
b
,
c
are positive integers, prove that for each positive integer
k
k
k
,
n
2
k
n^{2k}
n
2
k
can be expressed in the form
A
2
+
B
2
+
C
2
A^2 +B^2 +C^2
A
2
+
B
2
+
C
2
, where
A
,
B
,
C
A, B, C
A
,
B
,
C
are positive integers.
33
1
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P 33
Let
a
1
,
a
2
,
⋯
,
a
k
a_{1}, a_{2}, \cdots, a_{k}
a
1
,
a
2
,
⋯
,
a
k
be relatively prime positive integers. Determine the largest integer which cannot be expressed in the form
x
1
a
2
a
3
⋯
a
k
+
x
2
a
1
a
3
⋯
a
k
+
⋯
+
x
k
a
1
a
2
⋯
a
k
−
1
x_{1}a_{2}a_{3}\cdots a_{k}+x_{2}a_{1}a_{3}\cdots a_{k}+\cdots+x_{k}a_{1}a_{2}\cdots a_{k-1}
x
1
a
2
a
3
⋯
a
k
+
x
2
a
1
a
3
⋯
a
k
+
⋯
+
x
k
a
1
a
2
⋯
a
k
−
1
for some nonnegative integers
x
1
,
x
2
,
⋯
,
x
k
x_{1}, x_{2}, \cdots, x_{k}
x
1
,
x
2
,
⋯
,
x
k
.
32
1
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P 32
A composite positive integer is a product
a
b
ab
ab
with
a
a
a
and
b
b
b
not necessarily distinct integers in
{
2
,
3
,
4
,
…
}
\{2,3,4,\dots\}
{
2
,
3
,
4
,
…
}
. Show that every composite positive integer is expressible as
x
y
+
x
z
+
y
z
+
1
xy+xz+yz+1
x
y
+
x
z
+
yz
+
1
, with
x
,
y
,
z
x,y,z
x
,
y
,
z
positive integers.
31
1
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P 31
A finite sequence of integers
a
0
,
a
1
,
⋯
,
a
n
a_{0}, a_{1}, \cdots, a_{n}
a
0
,
a
1
,
⋯
,
a
n
is called quadratic if for each
i
∈
{
1
,
2
,
⋯
,
n
}
i \in \{1,2,\cdots,n \}
i
∈
{
1
,
2
,
⋯
,
n
}
we have the equality
∣
a
i
−
a
i
−
1
∣
=
i
2
\vert a_{i}-a_{i-1} \vert = i^2
∣
a
i
−
a
i
−
1
∣
=
i
2
. [*] Prove that for any two integers
b
b
b
and
c
c
c
, there exists a natural number
n
n
n
and a quadratic sequence with
a
0
=
b
a_{0}=b
a
0
=
b
and
a
n
=
c
a_{n}=c
a
n
=
c
. [*] Find the smallest natural number
n
n
n
for which there exists a quadratic sequence with
a
0
=
0
a_{0}=0
a
0
=
0
and
a
n
=
1996
a_{n}=1996
a
n
=
1996
.
30
1
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P 30
Let
a
1
,
a
2
,
a
3
,
⋯
a_{1}, a_{2}, a_{3}, \cdots
a
1
,
a
2
,
a
3
,
⋯
be an increasing sequence of nonnegative integers such that every nonnegative integer can be expressed uniquely in the form
a
i
+
2
a
j
+
4
a
k
a_{i}+2a_{j}+4a_{k}
a
i
+
2
a
j
+
4
a
k
, where
i
,
j
,
i, j,
i
,
j
,
and
k
k
k
are not necessarily distinct. Determine
a
1998
a_{1998}
a
1998
.
29
1
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P 29
Show that the set of positive integers which cannot be represented as a sum of distinct perfect squares is finite.
28
1
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P 28
Prove that any positive integer can be represented as a sum of Fibonacci numbers, no two of which are consecutive.
27
1
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P 27
Determine, with proof, the largest number which is the product of positive integers whose sum is
1976
1976
1976
.
26
1
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P 26
Let
a
,
b
a, b
a
,
b
and
c
c
c
be positive integers, no two of which have a common divisor greater than
1
1
1
. Show that
2
a
b
c
−
a
b
−
b
c
−
c
a
2abc-ab-bc-ca
2
ab
c
−
ab
−
b
c
−
c
a
is the largest integer which cannot be expressed in the form
x
b
c
+
y
c
a
+
z
a
b
xbc+yca+zab
x
b
c
+
yc
a
+
z
ab
, where
x
,
y
,
z
∈
N
0
x, y, z \in \mathbb{N}_{0}
x
,
y
,
z
∈
N
0
25
1
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P 25
Let
a
a
a
and
b
b
b
be positive integers with
gcd
(
a
,
b
)
=
1
\gcd(a, b)=1
g
cd
(
a
,
b
)
=
1
. Show that every integer greater than
a
b
−
a
−
b
ab-a-b
ab
−
a
−
b
can be expressed in the form
a
x
+
b
y
ax+by
a
x
+
b
y
, where
x
,
y
∈
N
0
x, y \in \mathbb{N}_{0}
x
,
y
∈
N
0
.
24
1
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P 24
Show that any integer can be expressed as the form
a
2
+
b
2
−
c
2
a^{2}+b^{2}-c^{2}
a
2
+
b
2
−
c
2
, where
a
,
b
,
c
∈
Z
a, b, c \in \mathbb{Z}
a
,
b
,
c
∈
Z
.
23
1
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P 23
Show that there are infinitely many positive integers which cannot be expressed as the sum of squares.
22
1
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P 22
Show that an integer can be expressed as the difference of two squares if and only if it is not of the form
4
k
+
2
(
k
∈
Z
)
4k+2 \; (k \in \mathbb{Z})
4
k
+
2
(
k
∈
Z
)
.
21
1
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P 21
Let
A
A
A
be the set of positive integers of the form
a
2
+
2
b
2
a^2 +2b^2
a
2
+
2
b
2
, where
a
a
a
and
b
b
b
are integers and
b
≠
0
b \neq 0
b
=
0
. Show that if
p
p
p
is a prime number and
p
2
∈
A
p^2 \in A
p
2
∈
A
, then
p
∈
A
p \in A
p
∈
A
.
20
1
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P 20
If an integer
n
n
n
is such that
7
n
7n
7
n
is the form
a
2
+
3
b
2
a^2 +3b^2
a
2
+
3
b
2
, prove that
n
n
n
is also of that form.
19
1
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P 19
Let
n
n
n
be an integer of the form
a
2
+
b
2
a^2 + b^2
a
2
+
b
2
, where
a
a
a
and
b
b
b
are relatively prime integers and such that if
p
p
p
is a prime,
p
≤
n
p \leq \sqrt{n}
p
≤
n
, then
p
p
p
divides
a
b
ab
ab
. Determine all such
n
n
n
.
18
1
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P 18
Let
p
p
p
be a prime with
p
≡
1
(
m
o
d
4
)
p \equiv 1 \pmod{4}
p
≡
1
(
mod
4
)
. Let
a
a
a
be the unique integer such that
p
=
a
2
+
b
2
,
a
≡
−
1
(
m
o
d
4
)
,
b
≡
0
(
m
o
d
2
)
p=a^{2}+b^{2}, \; a \equiv-1 \pmod{4}, \; b \equiv 0 \; \pmod{2}
p
=
a
2
+
b
2
,
a
≡
−
1
(
mod
4
)
,
b
≡
0
(
mod
2
)
Prove that
∑
i
=
0
p
−
1
(
i
3
+
6
i
2
+
i
p
)
=
2
(
2
p
)
,
\sum^{p-1}_{i=0}\left( \frac{i^{3}+6i^{2}+i }{p}\right) = 2 \left( \frac{2}{p}\right),
i
=
0
∑
p
−
1
(
p
i
3
+
6
i
2
+
i
)
=
2
(
p
2
)
,
where
(
k
p
)
\left(\frac{k}{p}\right)
(
p
k
)
denotes the Legendre Symbol.
17
1
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P 17
Let
p
p
p
be a prime number of the form
4
k
+
1
4k+1
4
k
+
1
. Suppose that
r
r
r
is a quadratic residue of
p
p
p
and that
s
s
s
is a quadratic nonresidue of
p
p
p
. Show that
p
=
a
2
+
b
2
p=a^{2}+b^{2}
p
=
a
2
+
b
2
, where
a
=
1
2
∑
i
=
1
p
−
1
(
i
(
i
2
−
r
)
p
)
,
b
=
1
2
∑
i
=
1
p
−
1
(
i
(
i
2
−
s
)
p
)
.
a=\frac{1}{2}\sum^{p-1}_{i=1}\left( \frac{i(i^{2}-r)}{p}\right), b=\frac{1}{2}\sum^{p-1}_{i=1}\left( \frac{i(i^{2}-s)}{p}\right).
a
=
2
1
i
=
1
∑
p
−
1
(
p
i
(
i
2
−
r
)
)
,
b
=
2
1
i
=
1
∑
p
−
1
(
p
i
(
i
2
−
s
)
)
.
Here,
(
k
p
)
\left( \frac{k}{p}\right)
(
p
k
)
denotes the Legendre Symbol.
16
1
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P 16
Prove that there exist infinitely many integers
n
n
n
such that
n
,
n
+
1
,
n
+
2
n, n+1, n+2
n
,
n
+
1
,
n
+
2
are each the sum of the squares of two integers.
15
1
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P 15
Find all integers
m
>
1
m>1
m
>
1
such that
m
3
m^3
m
3
is a sum of
m
m
m
squares of consecutive integers.
14
1
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P 14
Let
n
n
n
be a non-negative integer. Find all non-negative integers
a
a
a
,
b
b
b
,
c
c
c
,
d
d
d
such that
a
2
+
b
2
+
c
2
+
d
2
=
7
⋅
4
n
.
a^{2}+b^{2}+c^{2}+d^{2}= 7 \cdot 4^{n}.
a
2
+
b
2
+
c
2
+
d
2
=
7
⋅
4
n
.
13
1
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P 13
Let
a
1
=
1
a_{1}=1
a
1
=
1
,
a
2
=
2
a_{2}=2
a
2
=
2
,
a
3
a_{3}
a
3
,
a
4
a_{4}
a
4
,
⋯
\cdots
⋯
be the sequence of positive integers of the form
2
α
3
β
2^{\alpha}3^{\beta}
2
α
3
β
, where
α
\alpha
α
and
β
\beta
β
are nonnegative integers. Prove that every positive integer is expressible in the form
a
i
1
+
a
i
2
+
⋯
+
a
i
n
,
a_{i_{1}}+a_{i_{2}}+\cdots+a_{i_{n}},
a
i
1
+
a
i
2
+
⋯
+
a
i
n
,
where no summand is a multiple of any other.
12
1
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P 12
The positive function
p
(
n
)
p(n)
p
(
n
)
is defined as the number of ways that the positive integer
n
n
n
can be written as a sum of positive integers. Show that, for all positive integers
n
≥
2
n \ge 2
n
≥
2
,
2
⌊
n
⌋
<
p
(
n
)
<
n
3
⌊
n
⌋
.
2^{\lfloor \sqrt{n}\rfloor}< p(n) < n^{3 \lfloor\sqrt{n}\rfloor }.
2
⌊
n
⌋
<
p
(
n
)
<
n
3
⌊
n
⌋
.
11
1
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P 11
For each positive integer
n
n
n
, let
f
(
n
)
f(n)
f
(
n
)
denote the number of ways of representing
n
n
n
as a sum of powers of 2 with nonnegative integer exponents. Representations which differ only in the ordering of their summands are considered to be the same. For instance,
f
(
4
)
=
4
f(4)=4
f
(
4
)
=
4
, because the number
4
4
4
can be represented in the following four ways:
4
,
2
+
2
,
2
+
1
+
1
,
1
+
1
+
1
+
1.
4, 2+2, 2+1+1, 1+1+1+1.
4
,
2
+
2
,
2
+
1
+
1
,
1
+
1
+
1
+
1.
Prove that, for any integer
n
≥
3
n \geq 3
n
≥
3
,
2
n
2
/
4
<
f
(
2
n
)
<
2
n
2
/
2
.
2^{n^{2}/4}< f(2^{n}) < 2^{n^{2}/2}.
2
n
2
/4
<
f
(
2
n
)
<
2
n
2
/2
.
10
1
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P 10
For each positive integer
n
,
S
(
n
)
\,n,\;S(n)\,
n
,
S
(
n
)
is defined to be the greatest integer such that, for every positive integer
k
≤
S
(
n
)
,
n
2
\,k\leq S(n),\;n^{2}\,
k
≤
S
(
n
)
,
n
2
can be written as the sum of
k
\,k\,
k
positive squares. [*] Prove that
S
(
n
)
≤
n
2
−
14
S(n)\leq n^{2}-14
S
(
n
)
≤
n
2
−
14
for each
n
≥
4
n\geq 4
n
≥
4
. [*] Find an integer
n
n
n
such that
S
(
n
)
=
n
2
−
14
S(n)=n^{2}-14
S
(
n
)
=
n
2
−
14
. [*] Prove that there are infinitely many integers
n
n
n
such that
S
(
n
)
=
n
2
−
14
S(n)=n^{2}-14
S
(
n
)
=
n
2
−
14
.
9
1
Hide problems
P 9
The integer
9
9
9
can be written as a sum of two consecutive integers: 9=4+5. Moreover it can be written as a sum of (more than one) consecutive positive integers in exactly two ways, namely 9=4+5= 2+3+4. Is there an integer which can be written as a sum of
1990
1990
1990
consecutive integers and which can be written as a sum of (more than one) consecutive positive integers in exactly
1990
1990
1990
ways?
8
1
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P 8
Prove that any positive integer can be represented as an aggregate of different powers of
3
3
3
, the terms in the aggregate being combined by the signs
+
+
+
and
−
-
−
appropriately chosen.
7
1
Hide problems
P 7
Prove that every integer
n
≥
12
n \ge 12
n
≥
12
is the sum of two composite numbers.
6
1
Hide problems
P 6
Show that every integer greater than
1
1
1
can be written as a sum of two square-free integers.
5
1
Hide problems
P 5
Show that any positive rational number can be represented as the sum of three positive rational cubes.
4
1
Hide problems
P 4
Determine all positive integers that are expressible in the form
a
2
+
b
2
+
c
2
+
c
,
a^{2}+b^{2}+c^{2}+c,
a
2
+
b
2
+
c
2
+
c
,
where
a
a
a
,
b
b
b
,
c
c
c
are integers.
3
1
Hide problems
P 3
Prove that infinitely many positive integers cannot be written in the form
x
1
3
+
x
2
5
+
x
3
7
+
x
4
9
+
x
5
11
,
{x_{1}}^{3}+{x_{2}}^{5}+{x_{3}}^{7}+{x_{4}}^{9}+{x_{5}}^{11},
x
1
3
+
x
2
5
+
x
3
7
+
x
4
9
+
x
5
11
,
where
x
1
,
x
2
,
x
3
,
x
4
,
x
5
∈
N
x_{1}, x_{2}, x_{3}, x_{4}, x_{5}\in \mathbb{N}
x
1
,
x
2
,
x
3
,
x
4
,
x
5
∈
N
.
2
1
Hide problems
P 2
Show that each integer
n
n
n
can be written as the sum of five perfect cubes (not necessarily positive).
1
1
Hide problems
P 1
Show that any integer can be expressed as a sum of two squares and a cube.