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Contests
National and Regional Contests
PEN Problems
PEN L Problems
PEN L Problems
Part of
PEN Problems
Subcontests
(13)
13
1
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L 13
The sequence
{
x
n
}
n
≥
1
\{x_{n}\}_{n \ge 1}
{
x
n
}
n
≥
1
is defined by
x
1
=
x
2
=
1
,
x
n
+
2
=
14
x
n
+
1
−
x
n
−
4.
x_{1}=x_{2}=1, \; x_{n+2}= 14x_{n+1}-x_{n}-4.
x
1
=
x
2
=
1
,
x
n
+
2
=
14
x
n
+
1
−
x
n
−
4.
Prove that
x
n
x_{n}
x
n
is always a perfect square.
12
1
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L 12
The sequence
{
a
n
}
n
≥
1
\{a_{n}\}_{n \ge 1}
{
a
n
}
n
≥
1
is defined by
a
1
=
1
,
a
2
=
12
,
a
3
=
20
,
a
n
+
3
=
2
a
n
+
2
+
2
a
n
+
1
−
a
n
.
a_{1}=1, \; a_{2}=12, \; a_{3}=20, \; a_{n+3}= 2a_{n+2}+2a_{n+1}-a_{n}.
a
1
=
1
,
a
2
=
12
,
a
3
=
20
,
a
n
+
3
=
2
a
n
+
2
+
2
a
n
+
1
−
a
n
.
Prove that
1
+
4
a
n
a
n
+
1
1+4a_{n}a_{n+1}
1
+
4
a
n
a
n
+
1
is a square for all
n
∈
N
n \in \mathbb{N}
n
∈
N
.
11
1
Hide problems
L 11
Let the sequence
{
K
n
}
n
≥
1
\{K_{n}\}_{n \ge 1}
{
K
n
}
n
≥
1
be defined by
K
1
=
2
,
K
2
=
8
,
K
n
+
2
=
3
K
n
+
1
−
K
n
+
5
(
−
1
)
n
.
K_{1}=2, K_{2}=8, K_{n+2}=3K_{n+1}-K_{n}+5(-1)^{n}.
K
1
=
2
,
K
2
=
8
,
K
n
+
2
=
3
K
n
+
1
−
K
n
+
5
(
−
1
)
n
.
Prove that if
K
n
K_{n}
K
n
is prime, then
n
n
n
must be a power of
3
3
3
.
10
1
Hide problems
L 10
The sequence
{
y
n
}
n
≥
1
\{y_{n}\}_{n \ge 1}
{
y
n
}
n
≥
1
is defined by
y
1
=
y
2
=
1
,
y
n
+
2
=
(
4
k
−
5
)
y
n
+
1
−
y
n
+
4
−
2
k
.
y_{1}=y_{2}=1,\;\; y_{n+2}= (4k-5)y_{n+1}-y_{n}+4-2k.
y
1
=
y
2
=
1
,
y
n
+
2
=
(
4
k
−
5
)
y
n
+
1
−
y
n
+
4
−
2
k
.
Determine all integers
k
k
k
such that each term of this sequence is a perfect square.
9
1
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L 9
Let
{
u
n
}
n
≥
0
\{u_{n}\}_{n \ge 0}
{
u
n
}
n
≥
0
be a sequence of positive integers defined by
u
0
=
1
,
u
n
+
1
=
a
u
n
+
b
,
u_{0}= 1, \;u_{n+1}= au_{n}+b,
u
0
=
1
,
u
n
+
1
=
a
u
n
+
b
,
where
a
,
b
∈
N
a, b \in \mathbb{N}
a
,
b
∈
N
. Prove that for any choice of
a
a
a
and
b
b
b
, the sequence
{
u
n
}
n
≥
0
\{u_{n}\}_{n \ge 0}
{
u
n
}
n
≥
0
contains infinitely many composite numbers.
8
1
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L 8
Let
{
x
n
}
n
≥
0
\{x_{n}\}_{n\ge0}
{
x
n
}
n
≥
0
and
{
y
n
}
n
≥
0
\{y_{n}\}_{n\ge0}
{
y
n
}
n
≥
0
be two sequences defined recursively as follows
x
0
=
1
,
x
1
=
4
,
x
n
+
2
=
3
x
n
+
1
−
x
n
,
x_{0}=1, \; x_{1}=4, \; x_{n+2}=3 x_{n+1}-x_{n},
x
0
=
1
,
x
1
=
4
,
x
n
+
2
=
3
x
n
+
1
−
x
n
,
y
0
=
1
,
y
1
=
2
,
y
n
+
2
=
3
y
n
+
1
−
y
n
.
y_{0}=1, \; y_{1}=2, \; y_{n+2}=3 y_{n+1}-y_{n}.
y
0
=
1
,
y
1
=
2
,
y
n
+
2
=
3
y
n
+
1
−
y
n
.
[*] Prove that
x
n
2
−
5
y
n
2
+
4
=
0
{x_{n}}^{2}-5{y_{n}}^{2}+4=0
x
n
2
−
5
y
n
2
+
4
=
0
for all non-negative integers. [*] Suppose that
a
a
a
,
b
b
b
are two positive integers such that
a
2
−
5
b
2
+
4
=
0
a^{2}-5b^{2}+4=0
a
2
−
5
b
2
+
4
=
0
. Prove that there exists a non-negative integer
k
k
k
such that
a
=
x
k
a=x_{k}
a
=
x
k
and
b
=
y
k
b=y_{k}
b
=
y
k
.
7
1
Hide problems
L 7
Let
m
m
m
be a positive integer. Define the sequence
{
a
n
}
n
≥
0
\{a_{n}\}_{n \ge 0}
{
a
n
}
n
≥
0
by
a
0
=
0
,
a
1
=
m
,
a
n
+
1
=
m
2
a
n
−
a
n
−
1
.
a_{0}=0, \; a_{1}=m, \; a_{n+1}=m^{2}a_{n}-a_{n-1}.
a
0
=
0
,
a
1
=
m
,
a
n
+
1
=
m
2
a
n
−
a
n
−
1
.
Prove that an ordered pair
(
a
,
b
)
(a, b)
(
a
,
b
)
of non-negative integers, with
a
≤
b
a \le b
a
≤
b
, gives a solution to the equation
a
2
+
b
2
a
b
+
1
=
m
2
\frac{a^{2}+b^{2}}{ab+1}= m^{2}
ab
+
1
a
2
+
b
2
=
m
2
if and only if
(
a
,
b
)
(a, b)
(
a
,
b
)
is of the form
(
a
n
,
a
n
+
1
)
(a_{n}, a_{n+1})
(
a
n
,
a
n
+
1
)
for some
n
≥
0
n \ge 0
n
≥
0
.
6
1
Hide problems
L 6
Prove that no Fibonacci number can be factored into a product of two smaller Fibonacci numbers, each greater than 1.
5
1
Hide problems
L 5
The Fibonacci sequence
{
F
n
}
\{F_{n}\}
{
F
n
}
is defined by
F
1
=
1
,
F
2
=
1
,
F
n
+
2
=
F
n
+
1
+
F
n
.
F_{1}=1, \; F_{2}=1, \; F_{n+2}=F_{n+1}+F_{n}.
F
1
=
1
,
F
2
=
1
,
F
n
+
2
=
F
n
+
1
+
F
n
.
Show that
F
2
n
−
1
2
+
F
2
n
+
1
2
+
1
=
3
F
2
n
−
1
F
2
n
+
1
F_{2n-1}^{2}+F_{2n+1}^{2}+1=3F_{2n-1}F_{2n+1}
F
2
n
−
1
2
+
F
2
n
+
1
2
+
1
=
3
F
2
n
−
1
F
2
n
+
1
for all
n
≥
1
n \ge 1
n
≥
1
.
4
1
Hide problems
L 4
The Fibonacci sequence
{
F
n
}
\{F_{n}\}
{
F
n
}
is defined by
F
1
=
1
,
F
2
=
1
,
F
n
+
2
=
F
n
+
1
+
F
n
.
F_{1}=1, \; F_{2}=1, \; F_{n+2}=F_{n+1}+F_{n}.
F
1
=
1
,
F
2
=
1
,
F
n
+
2
=
F
n
+
1
+
F
n
.
Show that
F
m
n
−
F
n
+
1
m
+
F
n
−
1
m
F_{mn}-F_{n+1}^{m}+F_{n-1}^{m}
F
mn
−
F
n
+
1
m
+
F
n
−
1
m
is divisible by
F
n
3
F_{n}^{3}
F
n
3
for all
m
≥
1
m \ge 1
m
≥
1
and
n
>
1
n>1
n
>
1
.
3
1
Hide problems
L 3
The Fibonacci sequence
{
F
n
}
\{F_{n}\}
{
F
n
}
is defined by
F
1
=
1
,
F
2
=
1
,
F
n
+
2
=
F
n
+
1
+
F
n
.
F_{1}=1, \; F_{2}=1, \; F_{n+2}=F_{n+1}+F_{n}.
F
1
=
1
,
F
2
=
1
,
F
n
+
2
=
F
n
+
1
+
F
n
.
Show that
F
m
n
−
1
−
F
n
−
1
m
F_{mn-1}-F_{n-1}^{m}
F
mn
−
1
−
F
n
−
1
m
is divisible by
F
n
2
F_{n}^{2}
F
n
2
for all
m
≥
1
m \ge 1
m
≥
1
and
n
>
1
n>1
n
>
1
.
2
1
Hide problems
L 2
The Fibonacci sequence
{
F
n
}
\{F_{n}\}
{
F
n
}
is defined by
F
1
=
1
,
F
2
=
1
,
F
n
+
2
=
F
n
+
1
+
F
n
.
F_{1}=1, \; F_{2}=1, \; F_{n+2}=F_{n+1}+F_{n}.
F
1
=
1
,
F
2
=
1
,
F
n
+
2
=
F
n
+
1
+
F
n
.
Show that
gcd
(
F
m
,
F
n
)
=
F
gcd
(
m
,
n
)
\gcd (F_{m}, F_{n})=F_{\gcd (m, n)}
g
cd
(
F
m
,
F
n
)
=
F
g
c
d
(
m
,
n
)
for all
m
,
n
∈
N
m, n \in \mathbb{N}
m
,
n
∈
N
.
1
1
Hide problems
L 1
An integer sequence
{
a
n
}
n
≥
1
\{a_{n}\}_{n \ge 1}
{
a
n
}
n
≥
1
is defined by
a
0
=
0
,
a
1
=
1
,
a
n
+
2
=
2
a
n
+
1
+
a
n
a_{0}=0, \; a_{1}=1, \; a_{n+2}=2a_{n+1}+a_{n}
a
0
=
0
,
a
1
=
1
,
a
n
+
2
=
2
a
n
+
1
+
a
n
Show that
2
k
2^{k}
2
k
divides
a
n
a_{n}
a
n
if and only if
2
k
2^{k}
2
k
divides
n
n
n
.