MathDB
Problems
Contests
National and Regional Contests
PEN Problems
PEN H Problems
PEN H Problems
Part of
PEN Problems
Subcontests
(90)
91
1
Hide problems
H 91
If
R
R
R
and
S
S
S
are two rectangles with integer sides such that the perimeter of
R
R
R
equals the area of
S
S
S
and the perimeter of
S
S
S
equals the area of
R
R
R
, then we call
R
R
R
and
S
S
S
a friendly pair of rectangles. Find all friendly pairs of rectangles.
90
1
Hide problems
H 90
Find all triples of positive integers
(
x
,
y
,
z
)
(x, y, z)
(
x
,
y
,
z
)
such that
(
x
+
y
)
(
1
+
x
y
)
=
2
z
.
(x+y)(1+xy)= 2^{z}.
(
x
+
y
)
(
1
+
x
y
)
=
2
z
.
89
1
Hide problems
H 89
Prove that the number
99999
+
111111
3
99999+111111\sqrt{3}
99999
+
111111
3
cannot be written in the form
(
A
+
B
3
)
2
(A+B\sqrt{3})^2
(
A
+
B
3
)
2
, where
A
A
A
and
B
B
B
are integers.
88
1
Hide problems
H 88
(Leo Moser) Show that the Diophantine equation
1
x
1
+
1
x
2
+
⋯
+
1
x
n
+
1
x
1
x
2
⋯
x
n
=
1
\frac{1}{x_{1}}+\frac{1}{x_{2}}+\cdots+\frac{1}{x_{n}}+\frac{1}{x_{1}x_{2}\cdots x_{n}}= 1
x
1
1
+
x
2
1
+
⋯
+
x
n
1
+
x
1
x
2
⋯
x
n
1
=
1
has at least one solution for every positive integers
n
n
n
.
87
1
Hide problems
H 87
What is the smallest perfect square that ends in
9009
9009
9009
?
86
1
Hide problems
H 86
A triangle with integer sides is called Heronian if its area is an integer. Does there exist a Heronian triangle whose sides are the arithmetic, geometric and harmonic means of two positive integers?
85
1
Hide problems
H 85
Find all integer solutions to
2
(
x
5
+
y
5
+
1
)
=
5
x
y
(
x
2
+
y
2
+
1
)
2(x^5 +y^5 +1)=5xy(x^2 +y^2 +1)
2
(
x
5
+
y
5
+
1
)
=
5
x
y
(
x
2
+
y
2
+
1
)
.
84
1
Hide problems
H 84
For what positive numbers
a
a
a
is
2
+
a
3
+
2
−
a
3
\sqrt[3]{2+\sqrt{a}}+\sqrt[3]{2-\sqrt{a}}
3
2
+
a
+
3
2
−
a
an integer?
83
1
Hide problems
H 83
Find all pairs
(
a
,
b
)
(a, b)
(
a
,
b
)
of positive integers such that
(
a
3
+
b
3
−
1
)
2
=
49
+
20
6
3
.
(\sqrt[3]{a}+\sqrt[3]{b}-1 )^{2}= 49+20 \sqrt[3]{6}.
(
3
a
+
3
b
−
1
)
2
=
49
+
20
3
6
.
82
1
Hide problems
H 82
Find all triples
(
a
,
b
,
c
)
(a, b, c)
(
a
,
b
,
c
)
of positive integers to the equation
a
!
b
!
=
a
!
+
b
!
+
c
!
.
a! b! = a!+b!+c!.
a
!
b
!
=
a
!
+
b
!
+
c
!
.
81
1
Hide problems
H 81
Find a pair of relatively prime four digit natural numbers
A
A
A
and
B
B
B
such that for all natural numbers
m
m
m
and
n
n
n
,
∣
A
m
−
B
n
∣
≥
400
\vert A^m -B^n \vert \ge 400
∣
A
m
−
B
n
∣
≥
400
.
80
1
Hide problems
H 80
Prove that if
a
,
b
,
c
,
d
a, b, c, d
a
,
b
,
c
,
d
are integers such that
d
=
(
a
+
2
3
b
+
4
3
c
)
2
d=( a+\sqrt[3]{2}b+\sqrt[3]{4}c)^{2}
d
=
(
a
+
3
2
b
+
3
4
c
)
2
then
d
d
d
is a perfect square.
79
1
Hide problems
H 79
Find all positive integers
m
m
m
and
n
n
n
for which
1
!
+
2
!
+
3
!
+
⋯
+
n
!
=
m
2
1!+2!+3!+\cdots+n!=m^{2}
1
!
+
2
!
+
3
!
+
⋯
+
n
!
=
m
2
78
1
Hide problems
H 78
Let
x
,
y
x, y
x
,
y
, and
z
z
z
be integers with
z
>
1
z>1
z
>
1
. Show that
(
x
+
1
)
2
+
(
x
+
2
)
2
+
⋯
+
(
x
+
99
)
2
≠
y
z
.
(x+1)^{2}+(x+2)^{2}+\cdots+(x+99)^{2}\neq y^{z}.
(
x
+
1
)
2
+
(
x
+
2
)
2
+
⋯
+
(
x
+
99
)
2
=
y
z
.
77
1
Hide problems
H 77
Find all pairwise relatively prime positive integers
l
,
m
,
n
l, m, n
l
,
m
,
n
such that
(
l
+
m
+
n
)
(
1
l
+
1
m
+
1
n
)
(l+m+n)\left( \frac{1}{l}+\frac{1}{m}+\frac{1}{n}\right)
(
l
+
m
+
n
)
(
l
1
+
m
1
+
n
1
)
is an integer.
76
1
Hide problems
H 76
Find all pairs
(
m
,
n
)
(m,n)
(
m
,
n
)
of integers that satisfy the equation
(
m
−
n
)
2
=
4
m
n
m
+
n
−
1
.
(m-n)^{2}=\frac{4mn}{m+n-1}.
(
m
−
n
)
2
=
m
+
n
−
1
4
mn
.
75
1
Hide problems
H 75
Let
a
,
b
a,b
a
,
b
, and
x
x
x
be positive integers such that
x
a
+
b
=
a
b
b
x^{a+b}=a^b{b}
x
a
+
b
=
a
b
b
. Prove that
a
=
x
a=x
a
=
x
and
b
=
x
x
b=x^{x}
b
=
x
x
.
74
1
Hide problems
H 74
Find all pairs
(
a
,
b
)
(a,b)
(
a
,
b
)
of positive integers that satisfy the equation
a
a
a
=
b
b
.
a^{a^{a}}= b^{b}.
a
a
a
=
b
b
.
73
1
Hide problems
H 73
Find all pairs
(
a
,
b
)
(a,b)
(
a
,
b
)
of positive integers that satisfy the equation
a
b
2
=
b
a
.
a^{b^{2}}= b^{a}.
a
b
2
=
b
a
.
72
1
Hide problems
H 72
Find all pairs
(
x
,
y
)
(x, y)
(
x
,
y
)
of positive rational numbers such that
x
y
=
y
x
x^{y}=y^{x}
x
y
=
y
x
.
71
1
Hide problems
H 71
Let
n
n
n
be a positive integer. Prove that the equation
x
+
y
+
1
x
+
1
y
=
3
n
x+y+\frac{1}{x}+\frac{1}{y}=3n
x
+
y
+
x
1
+
y
1
=
3
n
does not have solutions in positive rational numbers.
70
1
Hide problems
H 70
Show that the equation
{
x
3
}
+
{
y
3
}
=
{
z
3
}
\{x^3\}+\{y^3\}=\{z^3\}
{
x
3
}
+
{
y
3
}
=
{
z
3
}
has infinitely many rational non-integer solutions.
69
1
Hide problems
H 69
Determine all positive rational numbers
r
≠
1
r \neq 1
r
=
1
such that
r
r
−
1
\sqrt[r-1]{r}
r
−
1
r
is rational.
68
1
Hide problems
H 68
Consider the system
x
+
y
=
z
+
u
,
x+y=z+u,
x
+
y
=
z
+
u
,
2
x
y
=
z
u
.
2xy=zu.
2
x
y
=
z
u
.
Find the greatest value of the real constant
m
m
m
such that
m
≤
x
y
m \le \frac{x}{y}
m
≤
y
x
for any positive integer solution
(
x
,
y
,
z
,
u
)
(x, y, z, u)
(
x
,
y
,
z
,
u
)
of the system, with
x
≥
y
x \ge y
x
≥
y
.
67
1
Hide problems
H 67
Is there a positive integer
m
m
m
such that the equation
1
a
+
1
b
+
1
c
+
1
a
b
c
=
m
a
+
b
+
c
\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{abc}= \frac{m}{a+b+c}
a
1
+
b
1
+
c
1
+
ab
c
1
=
a
+
b
+
c
m
has infinitely many solutions in positive integers
a
,
b
,
c
a, b, c \;
a
,
b
,
c
?
66
1
Hide problems
H 66
Let
b
b
b
be a positive integer. Determine all
2002
2002
2002
-tuples of non-negative integers
(
a
1
,
a
2
,
⋯
,
a
2002
)
(a_{1}, a_{2}, \cdots, a_{2002})
(
a
1
,
a
2
,
⋯
,
a
2002
)
satisfying
∑
j
=
1
2002
a
j
a
j
=
2002
b
b
.
\sum^{2002}_{j=1}{a_{j}}^{a_{j}}=2002{b}^{b}.
j
=
1
∑
2002
a
j
a
j
=
2002
b
b
.
65
1
Hide problems
H 65
Determine all pairs
(
x
,
y
)
(x, y)
(
x
,
y
)
of integers such that
(
19
a
+
b
)
18
+
(
a
+
b
)
18
+
(
19
b
+
a
)
18
(19a+b)^{18}+(a+b)^{18}+(19b+a)^{18}
(
19
a
+
b
)
18
+
(
a
+
b
)
18
+
(
19
b
+
a
)
18
is a nonzero perfect square.
64
1
Hide problems
H 64
Show that there is no positive integer
k
k
k
for which the equation
(
n
−
1
)
!
+
1
=
n
k
(n-1)!+1=n^{k}
(
n
−
1
)!
+
1
=
n
k
is true when
n
n
n
is greater than
5
5
5
.
63
1
Hide problems
H 63
Show that
∣
1
2
m
−
5
n
∣
≥
7
\vert 12^m -5^n\vert \ge 7
∣1
2
m
−
5
n
∣
≥
7
for all
m
,
n
∈
N
m, n \in \mathbb{N}
m
,
n
∈
N
.
62
1
Hide problems
H 62
Solve the equation
7
x
−
3
y
=
4
7^x -3^y =4
7
x
−
3
y
=
4
in positive integers.
61
1
Hide problems
H 61
Solve the equation
2
x
−
5
=
1
1
y
2^x -5 =11^{y}
2
x
−
5
=
1
1
y
in positive integers.
60
1
Hide problems
H 60
Show that the equation
x
7
+
y
7
=
1998
z
x^7 + y^7 = {1998}^z
x
7
+
y
7
=
1998
z
has no solution in positive integers.
59
1
Hide problems
H 59
Solve the equation
2
8
x
=
1
9
y
+
8
7
z
28^x =19^y +87^z
2
8
x
=
1
9
y
+
8
7
z
, where
x
,
y
,
z
x, y, z
x
,
y
,
z
are integers.
58
1
Hide problems
H 58
Solve in positive integers the equation
1
0
a
+
2
b
−
3
c
=
1997
10^{a}+2^{b}-3^{c}=1997
1
0
a
+
2
b
−
3
c
=
1997
.
57
1
Hide problems
H 57
Show that the equation
(
n
k
)
=
m
l
{n \choose k}=m^{l}
(
k
n
)
=
m
l
has no integral solution with
l
≥
2
l \ge 2
l
≥
2
and
4
≤
k
≤
n
−
4
4 \le k \le n-4
4
≤
k
≤
n
−
4
.
56
1
Hide problems
H 56
Prove that the equation
∏
c
y
c
(
x
1
−
x
2
)
=
∏
c
y
c
(
x
1
−
x
3
)
\prod_{cyc} (x_1-x_2)= \prod_{cyc} (x_1-x_3)
∏
cyc
(
x
1
−
x
2
)
=
∏
cyc
(
x
1
−
x
3
)
has a solution in natural numbers where all
x
i
x_i
x
i
are different.
55
1
Hide problems
H 55
Given that
34
!
=
95232799
c
d
96041408476186096435
a
b
00000
0
(
10
)
,
34! = 95232799cd96041408476186096435ab000000_{(10)},
34
!
=
95232799
c
d
96041408476186096435
ab
00000
0
(
10
)
,
determine the digits
a
,
b
,
c
a, b, c
a
,
b
,
c
, and
d
d
d
.
54
1
Hide problems
H 54
Show that the number of integral-sided right triangles whose ratio of area to semi-perimeter is
p
m
p^{m}
p
m
, where
p
p
p
is a prime and
m
m
m
is an integer, is
m
+
1
m+1
m
+
1
if
p
=
2
p=2
p
=
2
and
2
m
+
1
2m+1
2
m
+
1
if
p
≠
2
p \neq 2
p
=
2
.
53
1
Hide problems
H 53
Suppose that
a
,
b
a, b
a
,
b
, and
p
p
p
are integers such that
b
≡
1
(
m
o
d
4
)
b \equiv 1 \; \pmod{4}
b
≡
1
(
mod
4
)
,
p
≡
3
(
m
o
d
4
)
p \equiv 3 \; \pmod{4}
p
≡
3
(
mod
4
)
,
p
p
p
is prime, and if
q
q
q
is any prime divisor of
a
a
a
such that
q
≡
3
(
m
o
d
4
)
q \equiv 3 \; \pmod{4}
q
≡
3
(
mod
4
)
, then
q
p
∣
a
2
q^{p}\vert a^{2}
q
p
∣
a
2
and
p
p
p
does not divide
q
−
1
q-1
q
−
1
(if
q
=
p
q=p
q
=
p
, then also
q
∣
b
q \vert b
q
∣
b
). Show that the equation
x
2
+
4
a
2
=
y
p
−
b
p
x^{2}+4a^{2}= y^{p}-b^{p}
x
2
+
4
a
2
=
y
p
−
b
p
has no solutions in integers.
52
1
Hide problems
H 52
Do there exist two right-angled triangles with integer length sides that have the lengths of exactly two sides in common?
51
1
Hide problems
H 51
Prove that the product of five consecutive positive integers is never a perfect square.
50
1
Hide problems
H 50
Show that the equation
y
2
=
x
3
+
2
a
3
−
3
b
2
y^{2}=x^{3}+2a^{3}-3b^2
y
2
=
x
3
+
2
a
3
−
3
b
2
has no solution in integers if
a
b
≠
0
ab \neq 0
ab
=
0
,
a
≢
1
(
m
o
d
3
)
a \not\equiv 1 \; \pmod{3}
a
≡
1
(
mod
3
)
,
3
3
3
does not divide
b
b
b
,
a
a
a
is odd if
b
b
b
is even, and
p
=
t
2
+
27
u
2
p=t^2 +27u^2
p
=
t
2
+
27
u
2
has a solution in integers
t
,
u
t,u
t
,
u
if
p
∣
a
p \vert a
p
∣
a
and
p
≡
1
(
m
o
d
3
)
p \equiv 1 \; \pmod{3}
p
≡
1
(
mod
3
)
.
49
1
Hide problems
H 49
Show that the only solutions of the equation
x
3
−
3
x
y
2
−
y
3
=
1
x^{3}-3xy^2 -y^3 =1
x
3
−
3
x
y
2
−
y
3
=
1
are given by
(
x
,
y
)
=
(
1
,
0
)
,
(
0
,
−
1
)
,
(
−
1
,
1
)
,
(
1
,
−
3
)
,
(
−
3
,
2
)
,
(
2
,
1
)
(x,y)=(1,0),(0,-1),(-1,1),(1,-3),(-3,2),(2,1)
(
x
,
y
)
=
(
1
,
0
)
,
(
0
,
−
1
)
,
(
−
1
,
1
)
,
(
1
,
−
3
)
,
(
−
3
,
2
)
,
(
2
,
1
)
.
48
1
Hide problems
H 48
Solve the equation
x
2
+
7
=
2
n
x^2 +7=2^n
x
2
+
7
=
2
n
in integers.
47
1
Hide problems
H 47
Show that the equation
x
4
+
y
4
+
4
z
4
=
1
x^4 +y^4 +4z^4 =1
x
4
+
y
4
+
4
z
4
=
1
has infinitely many rational solutions.
46
1
Hide problems
H 46
Let
a
,
b
,
c
,
d
,
e
,
f
a, b, c, d, e, f
a
,
b
,
c
,
d
,
e
,
f
be integers such that
b
2
−
4
a
c
>
0
b^{2}-4ac>0
b
2
−
4
a
c
>
0
is not a perfect square and
4
a
c
f
+
b
d
e
−
a
e
2
−
c
d
2
−
f
b
2
≠
0
4acf+bde-ae^{2}-cd^{2}-fb^{2}\neq 0
4
a
c
f
+
b
d
e
−
a
e
2
−
c
d
2
−
f
b
2
=
0
. Let
f
(
x
,
y
)
=
a
x
2
+
b
x
y
+
c
y
2
+
d
x
+
e
y
+
f
f(x, y)=ax^{2}+bxy+cy^{2}+dx+ey+f
f
(
x
,
y
)
=
a
x
2
+
b
x
y
+
c
y
2
+
d
x
+
ey
+
f
Suppose that
f
(
x
,
y
)
=
0
f(x, y)=0
f
(
x
,
y
)
=
0
has an integral solution. Show that
f
(
x
,
y
)
=
0
f(x, y)=0
f
(
x
,
y
)
=
0
has infinitely many integral solutions.
45
1
Hide problems
H 45
Show that there cannot be four squares in arithmetical progression.
44
1
Hide problems
H 44
For all
n
∈
N
n \in \mathbb{N}
n
∈
N
, show that the number of integral solutions
(
x
,
y
)
(x, y)
(
x
,
y
)
of
x
2
+
x
y
+
y
2
=
n
x^{2}+xy+y^{2}=n
x
2
+
x
y
+
y
2
=
n
is finite and a multiple of
6
6
6
.
43
1
Hide problems
H 43
Find all solutions in integers of
x
3
+
2
y
3
=
4
z
3
x^{3}+2y^{3}=4z^{3}
x
3
+
2
y
3
=
4
z
3
.
42
1
Hide problems
H 42
Find all integers
a
a
a
for which
x
3
−
x
+
a
x^3 -x+a
x
3
−
x
+
a
has three integer roots.
41
1
Hide problems
H 41
Suppose that
A
=
1
,
2
,
A=1,2,
A
=
1
,
2
,
or
3
3
3
. Let
a
a
a
and
b
b
b
be relatively prime integers such that
a
2
+
A
b
2
=
s
3
a^{2}+Ab^2 =s^3
a
2
+
A
b
2
=
s
3
for some integer
s
s
s
. Then, there are integers
u
u
u
and
v
v
v
such that
s
=
u
2
+
A
v
2
s=u^2 +Av^2
s
=
u
2
+
A
v
2
,
a
=
u
3
−
3
A
v
u
2
a =u^3 - 3Avu^2
a
=
u
3
−
3
A
v
u
2
, and
b
=
3
u
2
v
−
A
v
3
b=3u^{2}v -Av^3
b
=
3
u
2
v
−
A
v
3
.
40
1
Hide problems
H 40
Determine all pairs of rational numbers
(
x
,
y
)
(x, y)
(
x
,
y
)
such that
x
3
+
y
3
=
x
2
+
y
2
.
x^{3}+y^{3}= x^{2}+y^{2}.
x
3
+
y
3
=
x
2
+
y
2
.
39
1
Hide problems
H 39
Let
A
,
B
,
C
,
D
,
E
A, B, C, D, E
A
,
B
,
C
,
D
,
E
be integers,
B
≠
0
B \neq 0
B
=
0
and
F
=
A
D
2
−
B
C
D
+
B
2
E
≠
0
F=AD^{2}-BCD+B^{2}E \neq 0
F
=
A
D
2
−
BC
D
+
B
2
E
=
0
. Prove that the number
N
N
N
of pairs of integers
(
x
,
y
)
(x, y)
(
x
,
y
)
such that
A
x
2
+
B
x
y
+
C
x
+
D
y
+
E
=
0
,
Ax^{2}+Bxy+Cx+Dy+E=0,
A
x
2
+
B
x
y
+
C
x
+
Dy
+
E
=
0
,
satisfies
N
≤
2
d
(
∣
F
∣
)
N \le 2 d( \vert F \vert )
N
≤
2
d
(
∣
F
∣
)
, where
d
(
n
)
d(n)
d
(
n
)
denotes the number of positive divisors of positive integer
n
n
n
.
38
1
Hide problems
H 38
Suppose that
p
p
p
is an odd prime such that
2
p
+
1
2p+1
2
p
+
1
is also prime. Show that the equation
x
p
+
2
y
p
+
5
z
p
=
0
x^{p}+2y^{p}+5z^{p}=0
x
p
+
2
y
p
+
5
z
p
=
0
has no solutions in integers other than
(
0
,
0
,
0
)
(0,0,0)
(
0
,
0
,
0
)
.
37
1
Hide problems
H 37
Prove that for each positive integer
n
n
n
there exist odd positive integers
x
n
x_n
x
n
and
y
n
y_n
y
n
such that
x
n
2
+
7
y
n
2
=
2
n
{x_{n}}^2 +7{y_{n}}^2 =2^n
x
n
2
+
7
y
n
2
=
2
n
.
36
1
Hide problems
H 36
Prove that the equation
a
2
+
b
2
=
c
2
+
3
a^2 +b^2 =c^2 +3
a
2
+
b
2
=
c
2
+
3
has infinitely many integer solutions
(
a
,
b
,
c
)
(a, b, c)
(
a
,
b
,
c
)
.
34
1
Hide problems
H 34
Are there integers
m
m
m
and
n
n
n
such that
5
m
2
−
6
m
n
+
7
n
2
=
1985
5m^2 -6mn+7n^2 =1985
5
m
2
−
6
mn
+
7
n
2
=
1985
?
33
1
Hide problems
H 33
Does there exist an integer such that its cube is equal to
3
n
2
+
3
n
+
7
3n^2 +3n+7
3
n
2
+
3
n
+
7
, where
n
n
n
is integer?
32
1
Hide problems
H 32
Let
n
n
n
be a natural number. Solve in whole numbers the equation
x
n
+
y
n
=
(
x
−
y
)
n
+
1
.
x^{n}+y^{n}=(x-y)^{n+1}.
x
n
+
y
n
=
(
x
−
y
)
n
+
1
.
31
1
Hide problems
H 31
Determine all integer solutions of the system
2
u
v
−
x
y
=
16
,
2uv-xy=16,
2
uv
−
x
y
=
16
,
x
v
−
y
u
=
12.
xv-yu=12.
xv
−
y
u
=
12.
30
1
Hide problems
H 30
Let
a
a
a
,
b
b
b
,
c
c
c
be given integers,
a
>
0
a>0
a
>
0
,
a
c
−
b
2
=
p
ac-b^2=p
a
c
−
b
2
=
p
a squarefree positive integer. Let
M
(
n
)
M(n)
M
(
n
)
denote the number of pairs of integers
(
x
,
y
)
(x, y)
(
x
,
y
)
for which
a
x
2
+
b
x
y
+
c
y
2
=
n
ax^2 +bxy+cy^2=n
a
x
2
+
b
x
y
+
c
y
2
=
n
. Prove that
M
(
n
)
M(n)
M
(
n
)
is finite and
M
(
n
)
=
M
(
p
k
⋅
n
)
M(n)=M(p^{k} \cdot n)
M
(
n
)
=
M
(
p
k
⋅
n
)
for every integer
k
≥
0
k \ge 0
k
≥
0
.
29
1
Hide problems
H 29
Find all pairs of integers
(
x
,
y
)
(x, y)
(
x
,
y
)
satisfying the equality
y
(
x
2
+
36
)
+
x
(
y
2
−
36
)
+
y
2
(
y
−
12
)
=
0.
y(x^{2}+36)+x(y^{2}-36)+y^{2}(y-12)=0.
y
(
x
2
+
36
)
+
x
(
y
2
−
36
)
+
y
2
(
y
−
12
)
=
0.
28
1
Hide problems
H 28
Let
a
,
b
,
c
a, b, c
a
,
b
,
c
be positive integers such that
a
a
a
and
b
b
b
are relatively prime and
c
c
c
is relatively prime either to
a
a
a
or
b
b
b
. Prove that there exist infinitely many triples
(
x
,
y
,
z
)
(x, y, z)
(
x
,
y
,
z
)
of distinct positive integers such that
x
a
+
y
b
=
z
c
.
x^{a}+y^{b}= z^{c}.
x
a
+
y
b
=
z
c
.
27
1
Hide problems
H 27
Prove that there exist infinitely many positive integers
n
n
n
such that
p
=
n
r
p=nr
p
=
n
r
, where
p
p
p
and
r
r
r
are respectively the semi-perimeter and the inradius of a triangle with integer side lengths.
26
1
Hide problems
H 26
Solve in integers the following equation
n
2002
=
m
(
m
+
n
)
(
m
+
2
n
)
⋯
(
m
+
2001
n
)
.
n^{2002}=m(m+n)(m+2n)\cdots(m+2001n).
n
2002
=
m
(
m
+
n
)
(
m
+
2
n
)
⋯
(
m
+
2001
n
)
.
25
1
Hide problems
H 25
What is the smallest positive integer
t
t
t
such that there exist integers
x
1
,
x
2
,
⋯
,
x
t
x_{1},x_{2}, \cdots, x_{t}
x
1
,
x
2
,
⋯
,
x
t
with
x
1
3
+
x
2
3
+
⋯
+
x
t
3
=
200
2
2002
?
{x_{1}}^{3}+{x_{2}}^{3}+\cdots+{x_{t}}^{3}=2002^{2002}\;\;?
x
1
3
+
x
2
3
+
⋯
+
x
t
3
=
200
2
2002
?
24
1
Hide problems
H 24
Prove that if
n
n
n
is a positive integer such that the equation
x
3
−
3
x
y
2
+
y
3
=
n
.
x^{3}-3xy^{2}+y^{3}=n.
x
3
−
3
x
y
2
+
y
3
=
n
.
has a solution in integers
(
x
,
y
)
,
(x,y),
(
x
,
y
)
,
then it has at least three such solutions. Show that the equation has no solutions in integers when
n
=
2891
n=2891
n
=
2891
.
23
1
Hide problems
H 23
Find all
(
x
,
y
,
z
)
∈
Z
3
(x,y,z) \in {\mathbb{Z}}^3
(
x
,
y
,
z
)
∈
Z
3
such that
x
3
+
y
3
+
z
3
=
x
+
y
+
z
=
3
x^{3}+y^{3}+z^{3}=x+y+z=3
x
3
+
y
3
+
z
3
=
x
+
y
+
z
=
3
.
22
1
Hide problems
H 22
Find all integers
a
,
b
,
c
,
x
,
y
,
z
a,b,c,x,y,z
a
,
b
,
c
,
x
,
y
,
z
such that
a
+
b
+
c
=
x
y
z
,
x
+
y
+
z
=
a
b
c
,
a
≥
b
≥
c
≥
1
,
x
≥
y
≥
z
≥
1.
a+b+c=xyz, \; x+y+z=abc, \; a \ge b \ge c \ge 1, \; x \ge y \ge z \ge 1.
a
+
b
+
c
=
x
yz
,
x
+
y
+
z
=
ab
c
,
a
≥
b
≥
c
≥
1
,
x
≥
y
≥
z
≥
1.
21
1
Hide problems
H 21
Prove that the equation
6
(
6
a
2
+
3
b
2
+
c
2
)
=
5
n
2
6(6a^{2}+3b^{2}+c^{2}) = 5n^{2}
6
(
6
a
2
+
3
b
2
+
c
2
)
=
5
n
2
has no solutions in integers except
a
=
b
=
c
=
n
=
0
a=b=c=n=0
a
=
b
=
c
=
n
=
0
.
20
1
Hide problems
H 20
Determine all positive integers
n
n
n
for which the equation
x
n
+
(
2
+
x
)
n
+
(
2
−
x
)
n
=
0
x^{n}+(2+x)^{n}+(2-x)^{n}= 0
x
n
+
(
2
+
x
)
n
+
(
2
−
x
)
n
=
0
has an integer as a solution.
19
1
Hide problems
H 19
Find all
(
x
,
y
,
z
,
n
)
∈
N
4
(x, y, z, n) \in {\mathbb{N}}^4
(
x
,
y
,
z
,
n
)
∈
N
4
such that
x
3
+
y
3
+
z
3
=
n
x
2
y
2
z
2
x^3 +y^3 +z^3 =nx^2 y^2 z^2
x
3
+
y
3
+
z
3
=
n
x
2
y
2
z
2
.
18
1
Hide problems
H 18
Determine all positive integer solutions
(
x
,
y
,
z
,
t
)
(x, y, z, t)
(
x
,
y
,
z
,
t
)
of the equation
(
x
+
y
)
(
y
+
z
)
(
z
+
x
)
=
x
y
z
t
(x+y)(y+z)(z+x)=xyzt
(
x
+
y
)
(
y
+
z
)
(
z
+
x
)
=
x
yz
t
for which
gcd
(
x
,
y
)
=
gcd
(
y
,
z
)
=
gcd
(
z
,
x
)
=
1
\gcd(x, y)=\gcd(y, z)=\gcd(z, x)=1
g
cd
(
x
,
y
)
=
g
cd
(
y
,
z
)
=
g
cd
(
z
,
x
)
=
1
.
17
1
Hide problems
H 17
Find all positive integers
n
n
n
for which the equation
a
+
b
+
c
+
d
=
n
a
b
c
d
a+b+c+d=n \sqrt{abcd}
a
+
b
+
c
+
d
=
n
ab
c
d
has a solution in positive integers.
16
1
Hide problems
H 16
Find all pairs
(
a
,
b
)
(a,b)
(
a
,
b
)
of different positive integers that satisfy the equation
W
(
a
)
=
W
(
b
)
W(a)=W(b)
W
(
a
)
=
W
(
b
)
, where
W
(
x
)
=
x
4
−
3
x
3
+
5
x
2
−
9
x
W(x)=x^{4}-3x^{3}+5x^{2}-9x
W
(
x
)
=
x
4
−
3
x
3
+
5
x
2
−
9
x
.
15
1
Hide problems
H 15
Prove that there are no integers
x
x
x
and
y
y
y
satisfying
x
2
=
y
5
−
4
x^{2}=y^{5}-4
x
2
=
y
5
−
4
.
14
1
Hide problems
H 14
Show that the equation
x
2
+
y
5
=
z
3
x^2 +y^5 =z^3
x
2
+
y
5
=
z
3
has infinitely many solutions in integers
x
,
y
,
z
x, y, z
x
,
y
,
z
for which
x
y
z
≠
0
xyz \neq 0
x
yz
=
0
.
13
1
Hide problems
H 13
Find all pairs
(
x
,
y
)
(x,y)
(
x
,
y
)
of positive integers that satisfy the equation
y
2
=
x
3
+
16.
y^{2}=x^{3}+16.
y
2
=
x
3
+
16.
12
1
Hide problems
H 12
Find all
(
x
,
y
,
z
)
∈
N
3
(x,y,z) \in {\mathbb{N}}^3
(
x
,
y
,
z
)
∈
N
3
such that
x
4
−
y
4
=
z
2
x^{4}-y^{4}=z^{2}
x
4
−
y
4
=
z
2
.
11
1
Hide problems
H 11
Find all
(
x
,
y
,
n
)
∈
N
3
(x,y,n) \in {\mathbb{N}}^3
(
x
,
y
,
n
)
∈
N
3
such that
gcd
(
x
,
n
+
1
)
=
1
\gcd(x, n+1)=1
g
cd
(
x
,
n
+
1
)
=
1
and
x
n
+
1
=
y
n
+
1
x^{n}+1=y^{n+1}
x
n
+
1
=
y
n
+
1
.
10
1
Hide problems
H 10
Prove that there are unique positive integers
a
a
a
and
n
n
n
such that
a
n
+
1
−
(
a
+
1
)
n
=
2001.
a^{n+1}-(a+1)^{n}= 2001.
a
n
+
1
−
(
a
+
1
)
n
=
2001.
9
1
Hide problems
H 9
Determine all integers
a
a
a
for which the equation
x
2
+
a
x
y
+
y
2
=
1
x^{2}+axy+y^{2}=1
x
2
+
a
x
y
+
y
2
=
1
has infinitely many distinct integer solutions
x
,
y
x, \;y
x
,
y
.
8
1
Hide problems
H 8
Show that the equation
x
3
+
y
3
+
z
3
+
t
3
=
1999
x^{3}+y^{3}+z^{3}+t^{3}=1999
x
3
+
y
3
+
z
3
+
t
3
=
1999
has infinitely many integral solutions.
7
1
Hide problems
H 7
Determine all pairs
(
x
,
y
)
(x,y)
(
x
,
y
)
of positive integers satisfying the equation
(
x
+
y
)
2
−
2
(
x
y
)
2
=
1.
(x+y)^{2}-2(xy)^{2}=1.
(
x
+
y
)
2
−
2
(
x
y
)
2
=
1.
6
1
Hide problems
H 6
Show that there are infinitely many pairs
(
x
,
y
)
(x, y)
(
x
,
y
)
of rational numbers such that
x
3
+
y
3
=
9
x^3 +y^3 =9
x
3
+
y
3
=
9
.
5
1
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H 5
Find all pairs
(
x
,
y
)
(x, y)
(
x
,
y
)
of rational numbers such that
y
2
=
x
3
−
3
x
+
2
y^2 =x^3 -3x+2
y
2
=
x
3
−
3
x
+
2
.
4
1
Hide problems
H 4
Find all pairs
(
x
,
y
)
(x, y)
(
x
,
y
)
of positive rational numbers such that
x
2
+
3
y
2
=
1
x^{2}+3y^{2}=1
x
2
+
3
y
2
=
1
.
3
1
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H 3
Does there exist a solution to the equation
x
2
+
y
2
+
z
2
+
u
2
+
v
2
=
x
y
z
u
v
−
65
x^{2}+y^{2}+z^{2}+u^{2}+v^{2}=xyzuv-65
x
2
+
y
2
+
z
2
+
u
2
+
v
2
=
x
yz
uv
−
65
in integers with
x
,
y
,
z
,
u
,
v
x, y, z, u, v
x
,
y
,
z
,
u
,
v
greater than
1998
1998
1998
?
2
1
Hide problems
H 2
The number
21982145917308330487013369
21982145917308330487013369
21982145917308330487013369
is the thirteenth power of a positive integer. Which positive integer?
1
1
Hide problems
H 1
One of Euler's conjectures was disproved in the
1980
1980
1980
s by three American Mathematicians when they showed that there is a positive integer
n
n
n
such that
n
5
=
13
3
5
+
11
0
5
+
8
4
5
+
2
7
5
.
n^{5}= 133^{5}+110^{5}+84^{5}+27^{5}.
n
5
=
13
3
5
+
11
0
5
+
8
4
5
+
2
7
5
.
Find the value of
n
n
n
.