MathDB
Problems
Contests
National and Regional Contests
PEN Problems
PEN A Problems
PEN A Problems
Part of
PEN Problems
Subcontests
(116)
118
1
Hide problems
A 118
Determine the highest power of
1980
1980
1980
which divides
(
1980
n
)
!
(
n
!
)
1980
.
\frac{(1980n)!}{(n!)^{1980}}.
(
n
!
)
1980
(
1980
n
)!
.
117
1
Hide problems
A 117
Find the smallest positive integer
n
n
n
such that
2
1989
∣
m
n
−
1
2^{1989}\; \vert \; m^{n}-1
2
1989
∣
m
n
−
1
for all odd positive integers
m
>
1
m>1
m
>
1
.
116
1
Hide problems
A 116
What is the smallest positive integer that consists base 10 of each of the ten digits, each used exactly once, and is divisible by each of the digits
2
2
2
through
9
9
9
?
115
1
Hide problems
A 115
Does there exist a
4
4
4
-digit integer (in decimal form) such that no replacement of three of its digits by any other three gives a multiple of
1992
1992
1992
?
114
1
Hide problems
A 114
What is the greatest common divisor of the set of numbers
{
16
n
+
10
n
−
1
∣
n
=
1
,
2
,
⋯
}
?
\{{16}^{n}+10n-1 \; \vert \; n=1,2,\cdots \}?
{
16
n
+
10
n
−
1
∣
n
=
1
,
2
,
⋯
}?
113
1
Hide problems
A 113
Find all triples
(
l
,
m
,
n
)
(l, m, n)
(
l
,
m
,
n
)
of distinct positive integers satisfying
gcd
(
l
,
m
)
2
=
l
+
m
,
gcd
(
m
,
n
)
2
=
m
+
n
,
and
gcd
(
n
,
l
)
2
=
n
+
l
.
{\gcd(l, m)}^{2}= l+m, \;{\gcd(m, n)}^{2}= m+n, \; \text{and}\;\;{\gcd(n, l)}^{2}= n+l.
g
cd
(
l
,
m
)
2
=
l
+
m
,
g
cd
(
m
,
n
)
2
=
m
+
n
,
and
g
cd
(
n
,
l
)
2
=
n
+
l
.
112
1
Hide problems
A 112
Prove that there exist infinitely many pairs
(
a
,
b
)
(a, b)
(
a
,
b
)
of relatively prime positive integers such that
a
2
−
5
b
and
b
2
−
5
a
\frac{a^{2}-5}{b}\;\; \text{and}\;\; \frac{b^{2}-5}{a}
b
a
2
−
5
and
a
b
2
−
5
are both positive integers.
111
1
Hide problems
A 111
Find all natural numbers
n
n
n
such that the number
n
(
n
+
1
)
(
n
+
2
)
(
n
+
3
)
n(n+1)(n+2)(n+3)
n
(
n
+
1
)
(
n
+
2
)
(
n
+
3
)
has exactly three different prime divisors.
110
1
Hide problems
A 110
For each positive integer
n
n
n
, write the sum
∑
m
=
1
n
1
/
m
\sum_{m=1}^n 1/m
∑
m
=
1
n
1/
m
in the form
p
n
/
q
n
p_n/q_n
p
n
/
q
n
, where
p
n
p_n
p
n
and
q
n
q_n
q
n
are relatively prime positive integers. Determine all
n
n
n
such that 5 does not divide
q
n
q_n
q
n
.
109
1
Hide problems
A 109
Find all positive integers
a
a
a
and
b
b
b
such that
a
2
+
b
b
2
−
a
and
b
2
+
a
a
2
−
b
\frac{a^{2}+b}{b^{2}-a}\text{ and }\frac{b^{2}+a}{a^{2}-b}
b
2
−
a
a
2
+
b
and
a
2
−
b
b
2
+
a
are both integers.
108
1
Hide problems
A 108
For each integer
n
>
1
n>1
n
>
1
, let
p
(
n
)
p(n)
p
(
n
)
denote the largest prime factor of
n
n
n
. Determine all triples
(
x
,
y
,
z
)
(x, y, z)
(
x
,
y
,
z
)
of distinct positive integers satisfying [*]
x
,
y
,
z
x, y, z
x
,
y
,
z
are in arithmetic progression, [*]
p
(
x
y
z
)
≤
3
p(xyz) \le 3
p
(
x
yz
)
≤
3
.
107
1
Hide problems
A 107
Find four positive integers, each not exceeding
70000
70000
70000
and each having more than
100
100
100
divisors.
106
1
Hide problems
A 106
Determine the least possible value of the natural number
n
n
n
such that
n
!
n!
n
!
ends in exactly
1987
1987
1987
zeros.
105
1
Hide problems
A 105
Find the smallest positive integer
n
n
n
such that [*]
n
n
n
has exactly
144
144
144
distinct positive divisors, [*] there are ten consecutive integers among the positive divisors of
n
n
n
.
104
1
Hide problems
A 104
A wobbly number is a positive integer whose
d
i
g
i
t
s
digits
d
i
g
i
t
s
in base
10
10
10
are alternatively non-zero and zero the units digit being non-zero. Determine all positive integers which do not divide any wobbly number.
102
1
Hide problems
A 102
Determine all three-digit numbers
N
N
N
having the property that
N
N
N
is divisible by
11
,
11,
11
,
and
N
11
\frac{N}{11}
11
N
is equal to the sum of the squares of the digits of
N
.
N.
N
.
101
1
Hide problems
A 101
Find all composite numbers
n
n
n
having the property that each proper divisor
d
d
d
of
n
n
n
has
n
−
20
≤
d
≤
n
−
12
n-20 \le d \le n-12
n
−
20
≤
d
≤
n
−
12
.
100
1
Hide problems
A 100
Find all positive integers
n
n
n
such that
n
n
n
has exactly
6
6
6
positive divisors
1
<
d
1
<
d
2
<
d
3
<
d
4
<
n
1<d_{1}<d_{2}<d_{3}<d_{4}<n
1
<
d
1
<
d
2
<
d
3
<
d
4
<
n
and
1
+
n
=
5
(
d
1
+
d
2
+
d
3
+
d
4
)
1+n=5(d_{1}+d_{2}+d_{3}+d_{4})
1
+
n
=
5
(
d
1
+
d
2
+
d
3
+
d
4
)
.
99
1
Hide problems
A 99
Let
n
≥
2
n \ge 2
n
≥
2
be a positive integer, with divisors
1
=
d
1
<
d
2
<
⋯
<
d
k
=
n
.
1=d_{1}< d_{2}< \cdots < d_{k}=n \;.
1
=
d
1
<
d
2
<
⋯
<
d
k
=
n
.
Prove that
d
1
d
2
+
d
2
d
3
+
⋯
+
d
k
−
1
d
k
d_{1}d_{2}+d_{2}d_{3}+\cdots+d_{k-1}d_{k}
d
1
d
2
+
d
2
d
3
+
⋯
+
d
k
−
1
d
k
is always less than
n
2
n^{2}
n
2
, and determine when it divides
n
2
n^{2}
n
2
.
98
1
Hide problems
A 98
Let
n
n
n
be a positive integer with
k
≥
22
k\ge22
k
≥
22
divisors
1
=
d
1
<
d
2
<
⋯
<
d
k
=
n
1=d_{1}< d_{2}< \cdots < d_{k}=n
1
=
d
1
<
d
2
<
⋯
<
d
k
=
n
, all different. Determine all
n
n
n
such that
d
7
2
+
d
10
2
=
(
n
d
22
)
2
.
{d_{7}}^{2}+{d_{10}}^{2}= \left( \frac{n}{d_{22}}\right)^{2}.
d
7
2
+
d
10
2
=
(
d
22
n
)
2
.
97
1
Hide problems
A 97
Suppose that
n
n
n
is a positive integer and let
d
1
<
d
2
<
d
3
<
d
4
d_{1}<d_{2}<d_{3}<d_{4}
d
1
<
d
2
<
d
3
<
d
4
be the four smallest positive integer divisors of
n
n
n
. Find all integers
n
n
n
such that
n
=
d
1
2
+
d
2
2
+
d
3
2
+
d
4
2
.
n={d_{1}}^{2}+{d_{2}}^{2}+{d_{3}}^{2}+{d_{4}}^{2}.
n
=
d
1
2
+
d
2
2
+
d
3
2
+
d
4
2
.
96
1
Hide problems
A 96
Find all positive integers
n
n
n
that have exactly
16
16
16
positive integral divisors
d
1
,
d
2
⋯
,
d
16
d_{1},d_{2} \cdots, d_{16}
d
1
,
d
2
⋯
,
d
16
such that
1
=
d
1
<
d
2
<
⋯
<
d
16
=
n
1=d_{1}<d_{2}<\cdots<d_{16}=n
1
=
d
1
<
d
2
<
⋯
<
d
16
=
n
,
d
6
=
18
d_6=18
d
6
=
18
, and
d
9
−
d
8
=
17
d_{9}-d_{8}=17
d
9
−
d
8
=
17
.
95
1
Hide problems
A 95
Suppose that
a
a
a
and
b
b
b
are natural numbers such that
p
=
b
4
2
a
−
b
2
a
+
b
p=\frac{b}{4}\sqrt{\frac{2a-b}{2a+b}}
p
=
4
b
2
a
+
b
2
a
−
b
is a prime number. What is the maximum possible value of
p
p
p
?
94
1
Hide problems
A 94
Find all
n
∈
N
n \in \mathbb{N}
n
∈
N
such that
3
n
−
n
3^{n}-n
3
n
−
n
is divisible by
17
17
17
.
93
1
Hide problems
A 93
Find the largest positive integer
n
n
n
such that
n
n
n
is divisible by all the positive integers less than
n
3
\sqrt[3]{n}
3
n
.
92
1
Hide problems
A 92
Let
a
a
a
and
b
b
b
be positive integers. When
a
2
+
b
2
a^{2}+b^{2}
a
2
+
b
2
is divided by
a
+
b
,
a+b,
a
+
b
,
the quotient is
q
q
q
and the remainder is
r
.
r.
r
.
Find all pairs
(
a
,
b
)
(a,b)
(
a
,
b
)
such that
q
2
+
r
=
1977
q^{2}+r=1977
q
2
+
r
=
1977
.
91
1
Hide problems
A 91
Determine all pairs
(
a
,
b
)
(a, b)
(
a
,
b
)
of positive integers such that
a
b
2
+
b
+
7
ab^2+b+7
a
b
2
+
b
+
7
divides
a
2
b
+
a
+
b
a^2 b+a+b
a
2
b
+
a
+
b
.
90
1
Hide problems
A 90
Determine all pairs
(
x
,
y
)
(x, y)
(
x
,
y
)
of positive integers with
y
∣
x
2
+
1
y \vert x^2 +1
y
∣
x
2
+
1
and
x
2
∣
y
3
+
1
x^2 \vert y^3 +1
x
2
∣
y
3
+
1
.
89
1
Hide problems
A 89
Determine all pairs
(
a
,
b
)
(a, b)
(
a
,
b
)
of integers for which
a
2
+
b
2
+
3
a^{2}+b^{2}+3
a
2
+
b
2
+
3
is divisible by
a
b
ab
ab
.
88
1
Hide problems
A 88
Find all positive integers
n
n
n
such that
9
n
−
1
9^{n}-1
9
n
−
1
is divisible by
7
n
7^n
7
n
.
87
1
Hide problems
A 87
Find all positive integers
n
n
n
such that
3
n
−
1
3^{n}-1
3
n
−
1
is divisible by
2
n
2^n
2
n
.
86
1
Hide problems
A 86
Find all positive integers
(
x
,
n
)
(x, n)
(
x
,
n
)
such that
x
n
+
2
n
+
1
x^{n}+2^{n}+1
x
n
+
2
n
+
1
divides
x
n
+
1
+
2
n
+
1
+
1
x^{n+1}+2^{n+1}+1
x
n
+
1
+
2
n
+
1
+
1
.
85
1
Hide problems
A 85
Find all
n
∈
N
n \in \mathbb{N}
n
∈
N
such that
2
n
−
1
2^{n-1}
2
n
−
1
divides
n
!
n!
n
!
.
84
1
Hide problems
A 84
Determine all
n
∈
N
n \in \mathbb{N}
n
∈
N
for which [*]
n
n
n
is not the square of any integer, [*]
⌊
n
⌋
3
\lfloor \sqrt{n}\rfloor ^3
⌊
n
⌋
3
divides
n
2
n^2
n
2
.
83
1
Hide problems
A 83
Find all
n
∈
N
n \in \mathbb{N}
n
∈
N
such that
⌊
n
⌋
\lfloor \sqrt{n}\rfloor
⌊
n
⌋
divides
n
n
n
.
82
1
Hide problems
A 82
Which integers can be represented as
(
x
+
y
+
z
)
2
x
y
z
\frac{(x+y+z)^{2}}{xyz}
x
yz
(
x
+
y
+
z
)
2
where
x
x
x
,
y
y
y
, and
z
z
z
are positive integers?
81
1
Hide problems
A 81
Determine all triples of positive integers
(
a
,
m
,
n
)
(a, m, n)
(
a
,
m
,
n
)
such that
a
m
+
1
a^m +1
a
m
+
1
divides
(
a
+
1
)
n
(a+1)^n
(
a
+
1
)
n
.
80
1
Hide problems
A 80
Find all pairs of positive integers
m
,
n
≥
3
m, n \ge 3
m
,
n
≥
3
for which there exist infinitely many positive integers
a
a
a
such that
a
m
+
a
−
1
a
n
+
a
2
−
1
\frac{a^{m}+a-1}{a^{n}+a^{2}-1}
a
n
+
a
2
−
1
a
m
+
a
−
1
is itself an integer.
79
1
Hide problems
A 79
Determine all pairs of integers
(
a
,
b
)
(a, b)
(
a
,
b
)
such that
a
2
2
a
b
2
−
b
3
+
1
\frac{a^{2}}{2ab^{2}-b^{3}+1}
2
a
b
2
−
b
3
+
1
a
2
is a positive integer.
78
1
Hide problems
A 78
Determine all ordered pairs
(
m
,
n
)
(m, n)
(
m
,
n
)
of positive integers such that
n
3
+
1
m
n
−
1
\frac{n^{3}+1}{mn-1}
mn
−
1
n
3
+
1
is an integer.
77
1
Hide problems
A 77
Find all positive integers, representable uniquely as
x
2
+
y
x
y
+
1
,
\frac{x^{2}+y}{xy+1},
x
y
+
1
x
2
+
y
,
where
x
x
x
and
y
y
y
are positive integers.
76
1
Hide problems
A 76
Find all integers
a
,
b
,
c
\,a,b,c\,
a
,
b
,
c
with
1
<
a
<
b
<
c
\,1<a<b<c\,
1
<
a
<
b
<
c
such that
(
a
−
1
)
(
b
−
1
)
(
c
−
1
)
is a divisor of
a
b
c
−
1.
(a-1)(b-1)(c-1)\hspace{0.2in}\text{is a divisor of}\hspace{0.2in}abc-1.
(
a
−
1
)
(
b
−
1
)
(
c
−
1
)
is a divisor of
ab
c
−
1.
75
1
Hide problems
A 75
Find all triples
(
a
,
b
,
c
)
(a,b,c)
(
a
,
b
,
c
)
of positive integers such that
2
c
−
1
2^{c}-1
2
c
−
1
divides
2
a
+
2
b
+
1
2^{a}+2^{b}+1
2
a
+
2
b
+
1
.
74
1
Hide problems
A 74
Find an integer
n
n
n
, where
100
≤
n
≤
1997
100 \leq n \leq 1997
100
≤
n
≤
1997
, such that
2
n
+
2
n
\frac{2^{n}+2}{n}
n
2
n
+
2
is also an integer.
73
1
Hide problems
A 73
Determine all pairs
(
n
,
p
)
(n,p)
(
n
,
p
)
of positive integers such that [*]
p
p
p
is a prime,
n
>
1
n>1
n
>
1
, [*]
(
p
−
1
)
n
+
1
(p-1)^{n} + 1
(
p
−
1
)
n
+
1
is divisible by
n
p
−
1
n^{p-1}
n
p
−
1
.
72
1
Hide problems
A 72
Determine all pairs
(
n
,
p
)
(n,p)
(
n
,
p
)
of nonnegative integers such that [*]
p
p
p
is a prime, [*]
n
<
2
p
n<2p
n
<
2
p
, [*]
(
p
−
1
)
n
+
1
(p-1)^{n} + 1
(
p
−
1
)
n
+
1
is divisible by
n
p
−
1
n^{p-1}
n
p
−
1
.
71
1
Hide problems
A 71
Determine all integers
n
>
1
n > 1
n
>
1
such that
2
n
+
1
n
2
\frac{2^{n}+1}{n^{2}}
n
2
2
n
+
1
is an integer.
70
1
Hide problems
A 70
Suppose that
m
=
n
q
m=nq
m
=
n
q
, where
n
n
n
and
q
q
q
are positive integers. Prove that the sum of binomial coefficients
∑
k
=
0
n
−
1
(
gcd
(
n
,
k
)
q
gcd
(
n
,
k
)
)
\sum_{k=0}^{n-1}{ \gcd(n, k)q \choose \gcd(n, k)}
k
=
0
∑
n
−
1
(
g
cd
(
n
,
k
)
g
cd
(
n
,
k
)
q
)
is divisible by
m
m
m
.
69
1
Hide problems
A 69
Prove that if the odd prime
p
p
p
divides
a
b
−
1
a^{b}-1
a
b
−
1
, where
a
a
a
and
b
b
b
are positive integers, then
p
p
p
appears to the same power in the prime factorization of
b
(
a
d
−
1
)
b(a^{d}-1)
b
(
a
d
−
1
)
, where
d
=
gcd
(
b
,
p
−
1
)
d=\gcd(b,p-1)
d
=
g
cd
(
b
,
p
−
1
)
.
68
1
Hide problems
A 68
Suppose that
S
=
{
a
1
,
⋯
,
a
r
}
S=\{a_{1}, \cdots, a_{r}\}
S
=
{
a
1
,
⋯
,
a
r
}
is a set of positive integers, and let
S
k
S_{k}
S
k
denote the set of subsets of
S
S
S
with
k
k
k
elements. Show that
lcm
(
a
1
,
⋯
,
a
r
)
=
∏
i
=
1
r
∏
s
∈
S
i
gcd
(
s
)
(
(
−
1
)
i
)
.
\text{lcm}(a_{1}, \cdots, a_{r})=\prod_{i=1}^{r}\prod_{s\in S_{i}}\gcd(s)^{\left((-1)^{i}\right)}.
lcm
(
a
1
,
⋯
,
a
r
)
=
i
=
1
∏
r
s
∈
S
i
∏
g
cd
(
s
)
(
(
−
1
)
i
)
.
67
1
Hide problems
A 67
Prove that
(
2
n
n
)
2n \choose n
(
n
2
n
)
is divisible by
n
+
1
n+1
n
+
1
.
66
1
Hide problems
A 66
(Four Number Theorem) Let
a
,
b
,
c
,
a, b, c,
a
,
b
,
c
,
and
d
d
d
be positive integers such that
a
b
=
c
d
ab=cd
ab
=
c
d
. Show that there exists positive integers
p
,
q
,
r
,
s
p, q, r,s
p
,
q
,
r
,
s
such that
a
=
p
q
,
b
=
r
s
,
c
=
p
s
,
d
=
q
r
.
a=pq, \;\; b=rs, \;\; c=ps, \;\; d=qr.
a
=
pq
,
b
=
rs
,
c
=
p
s
,
d
=
q
r
.
65
1
Hide problems
A 65
Clara computed the product of the first
n
n
n
positive integers and Valerid computed the product of the first
m
m
m
even positive integers, where
m
≥
2
m \ge 2
m
≥
2
. They got the same answer. Prove that one of them had made a mistake.
64
1
Hide problems
A 64
The last digit of the number
x
2
+
x
y
+
y
2
x^2 +xy+y^2
x
2
+
x
y
+
y
2
is zero (where
x
x
x
and
y
y
y
are positive integers). Prove that two last digits of this numbers are zeros.
63
1
Hide problems
A 63
There is a large pile of cards. On each card one of the numbers
1
1
1
,
2
2
2
,
⋯
\cdots
⋯
,
n
n
n
is written. It is known that the sum of all numbers of all the cards is equal to
k
⋅
n
!
k \cdot n!
k
⋅
n
!
for some integer
k
k
k
. Prove that it is possible to arrange cards into
k
k
k
stacks so that the sum of numbers written on the cards in each stack is equal to
n
!
n!
n
!
.
62
1
Hide problems
A 62
Let
p
(
n
)
p(n)
p
(
n
)
be the greatest odd divisor of
n
n
n
. Prove that
1
2
n
∑
k
=
1
2
n
p
(
k
)
k
>
2
3
.
\frac{1}{2^{n}}\sum_{k=1}^{2^{n}}\frac{p(k)}{k}> \frac{2}{3}.
2
n
1
k
=
1
∑
2
n
k
p
(
k
)
>
3
2
.
61
1
Hide problems
A 61
For any positive integer
n
>
1
n>1
n
>
1
, let
p
(
n
)
p(n)
p
(
n
)
be the greatest prime divisor of
n
n
n
. Prove that there are infinitely many positive integers
n
n
n
with
p
(
n
)
<
p
(
n
+
1
)
<
p
(
n
+
2
)
.
p(n)<p(n+1)<p(n+2).
p
(
n
)
<
p
(
n
+
1
)
<
p
(
n
+
2
)
.
60
1
Hide problems
A 60
Prove that there exist an infinite number of ordered pairs
(
a
,
b
)
(a,b)
(
a
,
b
)
of integers such that for every positive integer
t
t
t
, the number
a
t
+
b
at+b
a
t
+
b
is a triangular number if and only if
t
t
t
is a triangular number.
59
1
Hide problems
A 59
Suppose that
n
n
n
has (at least) two essentially distinct representations as a sum of two squares. Specifically, let
n
=
s
2
+
t
2
=
u
2
+
v
2
n=s^{2}+t^{2}=u^{2}+v^{2}
n
=
s
2
+
t
2
=
u
2
+
v
2
, where
s
≥
t
≥
0
s \ge t \ge 0
s
≥
t
≥
0
,
u
≥
v
≥
0
u \ge v \ge 0
u
≥
v
≥
0
, and
s
>
u
s>u
s
>
u
. Show that
gcd
(
s
u
−
t
v
,
n
)
\gcd(su-tv, n)
g
cd
(
s
u
−
t
v
,
n
)
is a proper divisor of
n
n
n
.
58
1
Hide problems
A 58
Let
k
≥
14
k\ge 14
k
≥
14
be an integer, and let
p
k
p_k
p
k
be the largest prime number which is strictly less than
k
k
k
. You may assume that
p
k
≥
3
k
4
p_k\ge \tfrac{3k}{4}
p
k
≥
4
3
k
. Let
n
n
n
be a composite integer. Prove that [*] if
n
=
2
p
k
n=2p_k
n
=
2
p
k
, then
n
n
n
does not divide
(
n
−
k
)
!
(n-k)!
(
n
−
k
)!
, [*] if
n
>
2
p
k
n>2p_k
n
>
2
p
k
, then
n
n
n
divides
(
n
−
k
)
!
(n-k)!
(
n
−
k
)!
.
57
1
Hide problems
A 57
Prove that for every
n
∈
N
n \in \mathbb{N}
n
∈
N
the following proposition holds:
7
∣
3
n
+
n
3
7|3^n +n^3
7∣
3
n
+
n
3
if and only if
7
∣
3
n
n
3
+
1
7|3^{n} n^3 +1
7∣
3
n
n
3
+
1
.
56
1
Hide problems
A 56
Let
a
,
b
a, b
a
,
b
, and
c
c
c
be integers such that
a
+
b
+
c
a+b+c
a
+
b
+
c
divides
a
2
+
b
2
+
c
2
a^2 +b^2 +c^2
a
2
+
b
2
+
c
2
. Prove that there are infinitely many positive integers
n
n
n
such that
a
+
b
+
c
a+b+c
a
+
b
+
c
divides
a
n
+
b
n
+
c
n
a^n +b^n +c^n
a
n
+
b
n
+
c
n
.
55
1
Hide problems
A 55
Show that for every natural number
n
n
n
the product
(
4
−
2
1
)
(
4
−
2
2
)
(
4
−
2
3
)
⋯
(
4
−
2
n
)
\left( 4-\frac{2}{1}\right) \left( 4-\frac{2}{2}\right) \left( 4-\frac{2}{3}\right) \cdots \left( 4-\frac{2}{n}\right)
(
4
−
1
2
)
(
4
−
2
2
)
(
4
−
3
2
)
⋯
(
4
−
n
2
)
is an integer.
54
1
Hide problems
A 54
A natural number
n
n
n
is said to have the property
P
P
P
, if whenever
n
n
n
divides
a
n
−
1
a^{n}-1
a
n
−
1
for some integer
a
a
a
,
n
2
n^2
n
2
also necessarily divides
a
n
−
1
a^{n}-1
a
n
−
1
. [*] Show that every prime number
n
n
n
has the property
P
P
P
. [*] Show that there are infinitely many composite numbers
n
n
n
that possess the property
P
P
P
.
53
1
Hide problems
A 53
Suppose that
x
,
y
,
x, y,
x
,
y
,
and
z
z
z
are positive integers with
x
y
=
z
2
+
1
xy=z^2 +1
x
y
=
z
2
+
1
. Prove that there exist integers
a
,
b
,
c
,
a, b, c,
a
,
b
,
c
,
and
d
d
d
such that
x
=
a
2
+
b
2
x=a^2 +b^2
x
=
a
2
+
b
2
,
y
=
c
2
+
d
2
y=c^2 +d^2
y
=
c
2
+
d
2
, and
z
=
a
c
+
b
d
z=ac+bd
z
=
a
c
+
b
d
.
52
1
Hide problems
A 52
Let
d
d
d
be any positive integer not equal to 2, 5, or 13. Show that one can find distinct
a
a
a
and
b
b
b
in the set
{
2
,
5
,
13
,
d
}
\{2,5,13,d\}
{
2
,
5
,
13
,
d
}
such that
a
b
−
1
ab - 1
ab
−
1
is not a perfect square.
51
1
Hide problems
A 51
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
and
d
d
d
be odd integers such that
0
<
a
<
b
<
c
<
d
0<a<b<c<d
0
<
a
<
b
<
c
<
d
and
a
d
=
b
c
ad=bc
a
d
=
b
c
. Prove that if
a
+
d
=
2
k
a+d=2^{k}
a
+
d
=
2
k
and
b
+
c
=
2
m
b+c=2^{m}
b
+
c
=
2
m
for some integers
k
k
k
and
m
m
m
, then
a
=
1
a=1
a
=
1
.
50
1
Hide problems
A 50
Show that every integer
k
>
1
k>1
k
>
1
has a multiple less than
k
4
k^4
k
4
whose decimal expansion has at most four distinct digits.
49
1
Hide problems
A 49
Prove that there is no positive integer
n
n
n
such that, for
k
=
1
,
2
,
⋯
,
9
,
k=1, 2, \cdots, 9,
k
=
1
,
2
,
⋯
,
9
,
the leftmost digit of
(
n
+
k
)
!
(n+k)!
(
n
+
k
)!
equals
k
k
k
.
48
1
Hide problems
A 48
Let
n
n
n
be a positive integer. Prove that
1
3
+
⋯
+
1
2
n
+
1
\frac{1}{3}+\cdots+\frac{1}{2n+1}
3
1
+
⋯
+
2
n
+
1
1
is not an integer.
47
1
Hide problems
A 47
Let
n
n
n
be a positive integer with
n
>
1
n>1
n
>
1
. Prove that
1
2
+
⋯
+
1
n
\frac{1}{2}+\cdots+\frac{1}{n}
2
1
+
⋯
+
n
1
is not an integer.
46
1
Hide problems
A 46
Let
a
a
a
and
b
b
b
be integers. Show that
a
a
a
and
b
b
b
have the same parity if and only if there exist integers
c
c
c
and
d
d
d
such that
a
2
+
b
2
+
c
2
+
1
=
d
2
a^2 +b^2 +c^2 +1 = d^2
a
2
+
b
2
+
c
2
+
1
=
d
2
.
45
1
Hide problems
A 45
Let
b
,
m
,
n
∈
N
b,m,n\in\mathbb{N}
b
,
m
,
n
∈
N
with
b
>
1
b>1
b
>
1
and
m
≠
n
m\not=n
m
=
n
. Suppose that
b
m
−
1
b^{m}-1
b
m
−
1
and
b
n
−
1
b^{n}-1
b
n
−
1
have the same set of prime divisors. Show that
b
+
1
b+1
b
+
1
must be a power of
2
2
2
.
44
1
Hide problems
A 44
Suppose that
4
n
+
2
n
+
1
4^{n}+2^{n}+1
4
n
+
2
n
+
1
is prime for some positive integer
n
n
n
. Show that
n
n
n
must be a power of
3
3
3
.
43
1
Hide problems
A 43
Suppose that
p
p
p
is a prime number and is greater than
3
3
3
. Prove that
7
p
−
6
p
−
1
7^{p}-6^{p}-1
7
p
−
6
p
−
1
is divisible by
43
43
43
.
42
1
Hide problems
A 42
Suppose that
2
n
+
1
2^n +1
2
n
+
1
is an odd prime for some positive integer
n
n
n
. Show that
n
n
n
must be a power of
2
2
2
.
41
1
Hide problems
A 41
Show that there are infinitely many composite numbers
n
n
n
such that
3
n
−
1
−
2
n
−
1
3^{n-1}-2^{n-1}
3
n
−
1
−
2
n
−
1
is divisible by
n
n
n
.
40
1
Hide problems
A 40
Determine the greatest common divisor of the elements of the set
{
n
13
−
n
∣
n
∈
Z
}
.
\{n^{13}-n \; \vert \; n \in \mathbb{Z}\}.
{
n
13
−
n
∣
n
∈
Z
}
.
39
1
Hide problems
A 39
Let
n
n
n
be a positive integer. Prove that the following two statements are equivalent. [*]
n
n
n
is not divisible by
4
4
4
[*] There exist
a
,
b
∈
Z
a, b \in \mathbb{Z}
a
,
b
∈
Z
such that
a
2
+
b
2
+
1
a^{2}+b^{2}+1
a
2
+
b
2
+
1
is divisible by
n
n
n
.
38
1
Hide problems
A 38
Let
p
p
p
be a prime with
p
>
5
p>5
p
>
5
, and let
S
=
{
p
−
n
2
∣
n
∈
N
,
n
2
<
p
}
S=\{p-n^2 \vert n \in \mathbb{N}, {n}^{2}<p \}
S
=
{
p
−
n
2
∣
n
∈
N
,
n
2
<
p
}
. Prove that
S
S
S
contains two elements
a
a
a
and
b
b
b
such that
a
∣
b
a \vert b
a
∣
b
and
1
<
a
<
b
1<a<b
1
<
a
<
b
.
37
1
Hide problems
A 37
If
n
n
n
is a natural number, prove that the number
(
n
+
1
)
(
n
+
2
)
⋯
(
n
+
10
)
(n+1)(n+2)\cdots(n+10)
(
n
+
1
)
(
n
+
2
)
⋯
(
n
+
10
)
is not a perfect square.
36
1
Hide problems
A 36
Let
n
n
n
and
q
q
q
be integers with
n
≥
5
n \ge 5
n
≥
5
,
2
≤
q
≤
n
2 \le q \le n
2
≤
q
≤
n
. Prove that
q
−
1
q-1
q
−
1
divides
⌊
(
n
−
1
)
!
q
⌋
\left\lfloor \frac{(n-1)!}{q}\right\rfloor
⌊
q
(
n
−
1
)!
⌋
.
35
1
Hide problems
A 35
Let
p
≥
5
p \ge 5
p
≥
5
be a prime number. Prove that there exists an integer
a
a
a
with
1
≤
a
≤
p
−
2
1 \le a \le p-2
1
≤
a
≤
p
−
2
such that neither
a
p
−
1
−
1
a^{p-1} -1
a
p
−
1
−
1
nor
(
a
+
1
)
p
−
1
−
1
(a+1)^{p-1} -1
(
a
+
1
)
p
−
1
−
1
is divisible by
p
2
p^2
p
2
.
34
1
Hide problems
A 34
Let
p
1
,
p
2
,
⋯
,
p
n
p_{1}, p_{2}, \cdots, p_{n}
p
1
,
p
2
,
⋯
,
p
n
be distinct primes greater than
3
3
3
. Show that
2
p
1
p
2
⋯
p
n
+
1
2^{p_{1}p_{2}\cdots p_{n}}+1
2
p
1
p
2
⋯
p
n
+
1
has at least
4
n
4^{n}
4
n
divisors.
33
1
Hide problems
A 33
Let
a
,
b
,
x
∈
N
a,b,x\in \mathbb{N}
a
,
b
,
x
∈
N
with
b
>
1
b>1
b
>
1
and such that
b
n
−
1
b^{n}-1
b
n
−
1
divides
a
a
a
. Show that in base
b
b
b
, the number
a
a
a
has at least
n
n
n
non-zero digits.
32
1
Hide problems
A 32
Let
a
a
a
and
b
b
b
be natural numbers such that
a
b
=
1
−
1
2
+
1
3
−
1
4
+
⋯
−
1
1318
+
1
1319
.
\frac{a}{b}=1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\cdots-\frac{1}{1318}+\frac{1}{1319}.
b
a
=
1
−
2
1
+
3
1
−
4
1
+
⋯
−
1318
1
+
1319
1
.
Prove that
a
a
a
is divisible by
1979
1979
1979
.
31
1
Hide problems
A 31
Show that there exist infinitely many positive integers
n
n
n
such that
n
2
+
1
n^{2}+1
n
2
+
1
divides
n
!
n!
n
!
.
30
1
Hide problems
A 30
Show that if
n
≥
6
n \ge 6
n
≥
6
is composite, then
n
n
n
divides
(
n
−
1
)
!
(n-1)!
(
n
−
1
)!
.
29
1
Hide problems
A 29
For which positive integers
k
k
k
, is it true that there are infinitely many pairs of positive integers
(
m
,
n
)
(m, n)
(
m
,
n
)
such that
(
m
+
n
−
k
)
!
m
!
n
!
\frac{(m+n-k)!}{m! \; n!}
m
!
n
!
(
m
+
n
−
k
)!
is an integer?
27
1
Hide problems
A 27
Show that the coefficients of a binomial expansion
(
a
+
b
)
n
(a+b)^n
(
a
+
b
)
n
where
n
n
n
is a positive integer, are all odd, if and only if
n
n
n
is of the form
2
k
−
1
2^{k}-1
2
k
−
1
for some positive integer
k
k
k
.
26
1
Hide problems
A 26
Let
m
m
m
and
n
n
n
be arbitrary non-negative integers. Prove that
(
2
m
)
!
(
2
n
)
!
m
!
n
!
(
m
+
n
)
!
\frac{(2m)!(2n)!}{m! n!(m+n)!}
m
!
n
!
(
m
+
n
)!
(
2
m
)!
(
2
n
)!
is an integer.
25
1
Hide problems
A 25
Show that
(
2
n
n
)
∣
lcm
(
1
,
2
,
⋯
,
2
n
)
{2n \choose n} \; \vert \; \text{lcm}(1,2, \cdots, 2n)
(
n
2
n
)
∣
lcm
(
1
,
2
,
⋯
,
2
n
)
for all positive integers
n
n
n
.
24
1
Hide problems
A 24
Let
p
>
3
p>3
p
>
3
is a prime number and
k
=
⌊
2
p
3
⌋
k=\lfloor\frac{2p}{3}\rfloor
k
=
⌊
3
2
p
⌋
. Prove that
(
p
1
)
+
(
p
2
)
+
⋯
+
(
p
k
)
{p \choose 1}+{p \choose 2}+\cdots+{p \choose k}
(
1
p
)
+
(
2
p
)
+
⋯
+
(
k
p
)
is divisible by
p
2
p^{2}
p
2
.
23
1
Hide problems
A 23
(Wolstenholme's Theorem) Prove that if
1
+
1
2
+
1
3
+
⋯
+
1
p
−
1
1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{p-1}
1
+
2
1
+
3
1
+
⋯
+
p
−
1
1
is expressed as a fraction, where
p
≥
5
p \ge 5
p
≥
5
is a prime, then
p
2
p^{2}
p
2
divides the numerator.
22
1
Hide problems
A 22
Prove that the number
∑
k
=
0
n
(
2
n
+
1
2
k
+
1
)
2
3
k
\sum_{k=0}^{n}\binom{2n+1}{2k+1}2^{3k}
k
=
0
∑
n
(
2
k
+
1
2
n
+
1
)
2
3
k
is not divisible by
5
5
5
for any integer
n
≥
0
n\geq 0
n
≥
0
.
21
1
Hide problems
A 21
Let n be a positive integer. Show that the product of
n
n
n
consecutive positive integers is divisible by
n
!
n!
n
!
20
1
Hide problems
A 20
Determine all positive integers
n
n
n
for which there exists an integer
m
m
m
such that
2
n
−
1
2^{n}-1
2
n
−
1
divides
m
2
+
9
m^{2}+9
m
2
+
9
.
19
1
Hide problems
A 19
Let
f
(
x
)
=
x
3
+
17
f(x)=x^3 +17
f
(
x
)
=
x
3
+
17
. Prove that for each natural number
n
≥
2
n \ge 2
n
≥
2
, there is a natural number
x
x
x
for which
f
(
x
)
f(x)
f
(
x
)
is divisible by
3
n
3^n
3
n
but not
3
n
+
1
3^{n+1}
3
n
+
1
.
18
1
Hide problems
A 18
Let
m
m
m
and
n
n
n
be natural numbers and let
m
n
+
1
mn+1
mn
+
1
be divisible by
24
24
24
. Show that
m
+
n
m+n
m
+
n
is divisible by
24
24
24
.
17
1
Hide problems
A 17
Let
m
m
m
and
n
n
n
be natural numbers such that
A
=
(
m
+
3
)
n
+
1
3
m
A=\frac{(m+3)^{n}+1}{3m}
A
=
3
m
(
m
+
3
)
n
+
1
is an integer. Prove that
A
A
A
is odd.
16
1
Hide problems
A 16
Determine if there exists a positive integer
n
n
n
such that
n
n
n
has exactly
2000
2000
2000
prime divisors and
2
n
+
1
2^{n}+1
2
n
+
1
is divisible by
n
n
n
.
15
1
Hide problems
A 15
Suppose that
k
≥
2
k \ge 2
k
≥
2
and
n
1
,
n
2
,
⋯
,
n
k
≥
1
n_{1}, n_{2}, \cdots, n_{k}\ge 1
n
1
,
n
2
,
⋯
,
n
k
≥
1
be natural numbers having the property
n
2
∣
2
n
1
−
1
,
n
3
∣
2
n
2
−
1
,
⋯
,
n
k
∣
2
n
k
−
1
−
1
,
n
1
∣
2
n
k
−
1.
n_{2}\; \vert \; 2^{n_{1}}-1, n_{3}\; \vert \; 2^{n_{2}}-1, \cdots, n_{k}\; \vert \; 2^{n_{k-1}}-1, n_{1}\; \vert \; 2^{n_{k}}-1.
n
2
∣
2
n
1
−
1
,
n
3
∣
2
n
2
−
1
,
⋯
,
n
k
∣
2
n
k
−
1
−
1
,
n
1
∣
2
n
k
−
1.
Show that
n
1
=
n
2
=
⋯
=
n
k
=
1
n_{1}=n_{2}=\cdots=n_{k}=1
n
1
=
n
2
=
⋯
=
n
k
=
1
.
14
1
Hide problems
A 14
Let
n
n
n
be an integer with
n
≥
2
n \ge 2
n
≥
2
. Show that
n
n
n
does not divide
2
n
−
1
2^{n}-1
2
n
−
1
.
13
1
Hide problems
A 13
Show that for all prime numbers
p
p
p
,
Q
(
p
)
=
∏
k
=
1
p
−
1
k
2
k
−
p
−
1
Q(p)=\prod^{p-1}_{k=1}k^{2k-p-1}
Q
(
p
)
=
k
=
1
∏
p
−
1
k
2
k
−
p
−
1
is an integer.
12
1
Hide problems
A 12
Let
k
,
m
,
k,m,
k
,
m
,
and
n
n
n
be natural numbers such that
m
+
k
+
1
m+k+1
m
+
k
+
1
is a prime greater than
n
+
1
n+1
n
+
1
. Let
c
s
=
s
(
s
+
1
)
.
c_{s}=s(s+1).
c
s
=
s
(
s
+
1
)
.
Prove that the product
(
c
m
+
1
−
c
k
)
(
c
m
+
2
−
c
k
)
⋯
(
c
m
+
n
−
c
k
)
(c_{m+1}-c_{k})(c_{m+2}-c_{k})\cdots (c_{m+n}-c_{k})
(
c
m
+
1
−
c
k
)
(
c
m
+
2
−
c
k
)
⋯
(
c
m
+
n
−
c
k
)
is divisible by the product
c
1
c
2
⋯
c
n
c_{1}c_{2}\cdots c_{n}
c
1
c
2
⋯
c
n
.
11
1
Hide problems
A 11
Let
a
,
b
,
c
,
d
a, b, c, d
a
,
b
,
c
,
d
be integers. Show that the product
(
a
−
b
)
(
a
−
c
)
(
a
−
d
)
(
b
−
c
)
(
b
−
d
)
(
c
−
d
)
(a-b)(a-c)(a-d)(b-c)(b-d)(c-d)
(
a
−
b
)
(
a
−
c
)
(
a
−
d
)
(
b
−
c
)
(
b
−
d
)
(
c
−
d
)
is divisible by
12
12
12
.
10
1
Hide problems
A 10
Let
n
n
n
be a positive integer with
n
≥
3
n \ge 3
n
≥
3
. Show that
n
n
n
n
−
n
n
n
n^{n^{n^{n}}}-n^{n^{n}}
n
n
n
n
−
n
n
n
is divisible by
1989
1989
1989
.
9
1
Hide problems
A 9
Prove that among any ten consecutive positive integers at least one is relatively prime to the product of the others.
8
1
Hide problems
A 8
The integers
a
a
a
and
b
b
b
have the property that for every nonnegative integer
n
n
n
the number of
2
n
a
+
b
2^n{a}+b
2
n
a
+
b
is the square of an integer. Show that
a
=
0
a=0
a
=
0
.
7
1
Hide problems
A 7
Let
n
n
n
be a positive integer such that
2
+
2
28
n
2
+
1
2+2\sqrt{28n^2 +1}
2
+
2
28
n
2
+
1
is an integer. Show that
2
+
2
28
n
2
+
1
2+2\sqrt{28n^2 +1}
2
+
2
28
n
2
+
1
is the square of an integer.
6
1
Hide problems
A 6
[*] Find infinitely many pairs of integers
a
a
a
and
b
b
b
with
1
<
a
<
b
1<a<b
1
<
a
<
b
, so that
a
b
ab
ab
exactly divides
a
2
+
b
2
−
1
a^{2}+b^{2}-1
a
2
+
b
2
−
1
. [*] With
a
a
a
and
b
b
b
as above, what are the possible values of
a
2
+
b
2
−
1
a
b
?
\frac{a^{2}+b^{2}-1}{ab}?
ab
a
2
+
b
2
−
1
?
5
1
Hide problems
A 5
Let
x
x
x
and
y
y
y
be positive integers such that
x
y
xy
x
y
divides
x
2
+
y
2
+
1
x^{2}+y^{2}+1
x
2
+
y
2
+
1
. Show that
x
2
+
y
2
+
1
x
y
=
3.
\frac{x^{2}+y^{2}+1}{xy}=3.
x
y
x
2
+
y
2
+
1
=
3.
4
1
Hide problems
A 4
If
a
,
b
,
c
a, b, c
a
,
b
,
c
are positive integers such that
0
<
a
2
+
b
2
−
a
b
c
≤
c
,
0 < a^{2}+b^{2}-abc \le c,
0
<
a
2
+
b
2
−
ab
c
≤
c
,
show that
a
2
+
b
2
−
a
b
c
a^{2}+b^{2}-abc
a
2
+
b
2
−
ab
c
is a perfect square.
3
1
Hide problems
A 3
Let
a
a
a
and
b
b
b
be positive integers such that
a
b
+
1
ab+1
ab
+
1
divides
a
2
+
b
2
a^{2}+b^{2}
a
2
+
b
2
. Show that
a
2
+
b
2
a
b
+
1
\frac{a^{2}+b^{2}}{ab+1}
ab
+
1
a
2
+
b
2
is the square of an integer.
2
1
Hide problems
A 2
Find infinitely many triples
(
a
,
b
,
c
)
(a, b, c)
(
a
,
b
,
c
)
of positive integers such that
a
a
a
,
b
b
b
,
c
c
c
are in arithmetic progression and such that
a
b
+
1
ab+1
ab
+
1
,
b
c
+
1
bc+1
b
c
+
1
, and
c
a
+
1
ca+1
c
a
+
1
are perfect squares.
1
1
Hide problems
A 1
Show that if
x
,
y
,
z
x, y, z
x
,
y
,
z
are positive integers, then
(
x
y
+
1
)
(
y
z
+
1
)
(
z
x
+
1
)
(xy+1)(yz+1)(zx+1)
(
x
y
+
1
)
(
yz
+
1
)
(
z
x
+
1
)
is a perfect square if and only if
x
y
+
1
xy+1
x
y
+
1
,
y
z
+
1
yz+1
yz
+
1
,
z
x
+
1
zx+1
z
x
+
1
are all perfect squares.