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Problems
Contests
National and Regional Contests
PEN Problems
PEN N Problems
PEN N Problems
Part of
PEN Problems
Subcontests
(16)
17
1
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N 17
Suppose that
a
a
a
and
b
b
b
are distinct real numbers such that
a
−
b
,
a
2
−
b
2
,
⋯
,
a
k
−
b
k
,
⋯
a-b, \; a^{2}-b^{2}, \; \cdots, \; a^{k}-b^{k}, \; \cdots
a
−
b
,
a
2
−
b
2
,
⋯
,
a
k
−
b
k
,
⋯
are all integers. Show that
a
a
a
and
b
b
b
are integers.
16
1
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N 16
Does there exist positive integers
a
1
<
a
2
<
⋯
<
a
100
a_{1}<a_{2}<\cdots<a_{100}
a
1
<
a
2
<
⋯
<
a
100
such that for
2
≤
k
≤
100
2 \le k \le 100
2
≤
k
≤
100
, the greatest common divisor of
a
k
−
1
a_{k-1}
a
k
−
1
and
a
k
a_{k}
a
k
is greater than the greatest common divisor of
a
k
a_{k}
a
k
and
a
k
+
1
a_{k+1}
a
k
+
1
?
15
1
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N 14
In the sequence
00
00
00
,
01
01
01
,
02
02
02
,
03
03
03
,
⋯
\cdots
⋯
,
99
99
99
the terms are rearranged so that each term is obtained from the previous one by increasing or decreasing one of its digits by
1
1
1
(for example,
29
29
29
can be followed by
19
19
19
,
39
39
39
, or
28
28
28
, but not by
30
30
30
or
20
20
20
). What is the maximal number of terms that could remain on their places?
13
1
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N 13
One member of an infinite arithmetic sequence in the set of natural numbers is a perfect square. Show that there are infinitely many members of this sequence having this property.
12
1
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N 12
The sequence
{
a
n
}
n
≥
1
\{a_{n}\}_{n \ge 1}
{
a
n
}
n
≥
1
is defined by
a
n
=
1
+
2
2
+
3
3
+
⋯
+
n
n
.
a_{n}= 1+2^{2}+3^{3}+\cdots+n^{n}.
a
n
=
1
+
2
2
+
3
3
+
⋯
+
n
n
.
Prove that there are infinitely many
n
n
n
such that
a
n
a_{n}
a
n
is composite.
11
1
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N 11
The infinite sequence of 2's and 3's
2
,
3
,
3
,
2
,
3
,
3
,
3
,
2
,
3
,
3
,
3
,
2
,
3
,
3
,
2
,
3
,
3
,
3
,
2
,
3
,
3
,
3
,
2
,
3
,
3
,
3
,
2
,
3
,
3
,
2
,
3
,
3
,
3
,
2
,
⋯
\begin{array}{l}2,3,3,2,3,3,3,2,3,3,3,2,3,3,2,3,3, \\ 3,2,3,3,3,2,3,3,3,2,3,3,2,3,3,3,2,\cdots \end{array}
2
,
3
,
3
,
2
,
3
,
3
,
3
,
2
,
3
,
3
,
3
,
2
,
3
,
3
,
2
,
3
,
3
,
3
,
2
,
3
,
3
,
3
,
2
,
3
,
3
,
3
,
2
,
3
,
3
,
2
,
3
,
3
,
3
,
2
,
⋯
has the property that, if one forms a second sequence that records the number of 3's between successive 2's, the result is identical to the given sequence. Show that there exists a real number
r
r
r
such that, for any
n
n
n
, the
n
n
n
th term of the sequence is 2 if and only if
n
=
1
+
⌊
r
m
⌋
n = 1+\lfloor rm \rfloor
n
=
1
+
⌊
r
m
⌋
for some nonnegative integer
m
m
m
.
10
1
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N 10
Let
a
,
b
a,b
a
,
b
be integers greater than 2. Prove that there exists a positive integer
k
k
k
and a finite sequence
n
1
,
n
2
,
…
,
n
k
n_1, n_2, \dots, n_k
n
1
,
n
2
,
…
,
n
k
of positive integers such that
n
1
=
a
n_1 = a
n
1
=
a
,
n
k
=
b
n_k = b
n
k
=
b
, and
n
i
n
i
+
1
n_i n_{i+1}
n
i
n
i
+
1
is divisible by
n
i
+
n
i
+
1
n_i + n_{i+1}
n
i
+
n
i
+
1
for each
i
i
i
(
1
≤
i
<
k
1 \leq i < k
1
≤
i
<
k
).
9
1
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N 9
Let
q
0
,
q
1
,
⋯
q_{0}, q_{1}, \cdots
q
0
,
q
1
,
⋯
be a sequence of integers such that a) for any
m
>
n
m > n
m
>
n
, m \minus{} n is a factor of q_{m} \minus{} q_{n}, b) item
∣
q
n
∣
≤
n
10
|q_n| \le n^{10}
∣
q
n
∣
≤
n
10
for all integers
n
≥
0
n \ge 0
n
≥
0
. Show that there exists a polynomial
Q
(
x
)
Q(x)
Q
(
x
)
satisfying q_{n} \equal{} Q(n) for all
n
n
n
.
8
1
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N 8
An integer sequence
{
a
n
}
n
≥
1
\{a_{n}\}_{n \ge 1}
{
a
n
}
n
≥
1
is given such that
2
n
=
∑
d
∣
n
a
d
2^{n}=\sum^{}_{d \vert n}a_{d}
2
n
=
d
∣
n
∑
a
d
for all
n
∈
N
n \in \mathbb{N}
n
∈
N
. Show that
a
n
a_{n}
a
n
is divisible by
n
n
n
for all
n
∈
N
n \in \mathbb{N}
n
∈
N
.
7
1
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N 7
Let
{
n
k
}
k
≥
1
\{n_{k}\}_{k \ge 1}
{
n
k
}
k
≥
1
be a sequence of natural numbers such that for
i
<
j
i<j
i
<
j
, the decimal representation of
n
i
n_{i}
n
i
does not occur as the leftmost digits of the decimal representation of
n
j
n_{j}
n
j
. Prove that
∑
k
=
1
∞
1
n
k
≤
1
1
+
1
2
+
⋯
+
1
9
.
\sum^{\infty}_{k=1}\frac{1}{n_{k}}\le \frac{1}{1}+\frac{1}{2}+\cdots+\frac{1}{9}.
k
=
1
∑
∞
n
k
1
≤
1
1
+
2
1
+
⋯
+
9
1
.
6
1
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N 6
Let
{
a
n
}
\{a_{n}\}
{
a
n
}
be a strictly increasing positive integers sequence such that
gcd
(
a
i
,
a
j
)
=
1
\gcd(a_{i}, a_{j})=1
g
cd
(
a
i
,
a
j
)
=
1
and
a
i
+
2
−
a
i
+
1
>
a
i
+
1
−
a
i
a_{i+2}-a_{i+1}>a_{i+1}-a_{i}
a
i
+
2
−
a
i
+
1
>
a
i
+
1
−
a
i
. Show that the infinite series
∑
i
=
1
∞
1
a
i
\sum^{\infty}_{i=1}\frac{1}{a_{i}}
i
=
1
∑
∞
a
i
1
converges.
5
1
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N 5
Prove that there exist two strictly increasing sequences
a
n
a_{n}
a
n
and
b
n
b_{n}
b
n
such that
a
n
(
a
n
+
1
)
a_{n}(a_{n} +1)
a
n
(
a
n
+
1
)
divides
b
n
2
+
1
b_{n}^2 +1
b
n
2
+
1
for every natural
n
n
n
.
4
1
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N 4
Show that if an infinite arithmetic progression of positive integers contains a square and a cube, it must contain a sixth power.
3
1
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N 3
Let
n
>
6
\,n>6\,
n
>
6
be an integer and
a
1
,
a
2
,
…
,
a
k
\,a_{1},a_{2},\ldots,a_{k}\,
a
1
,
a
2
,
…
,
a
k
be all the natural numbers less than
n
n
n
and relatively prime to
n
n
n
. If
a
2
−
a
1
=
a
3
−
a
2
=
⋯
=
a
k
−
a
k
−
1
>
0
,
a_{2}-a_{1}=a_{3}-a_{2}=\cdots =a_{k}-a_{k-1}>0,
a
2
−
a
1
=
a
3
−
a
2
=
⋯
=
a
k
−
a
k
−
1
>
0
,
prove that
n
\,n\,
n
must be either a prime number or a power of
2
\,2
2
.
2
1
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N 2
Let
a
n
a_{n}
a
n
be the last nonzero digit in the decimal representation of the number
n
!
n!
n
!
. Does the sequence
a
1
a_{1}
a
1
,
a
2
a_{2}
a
2
,
a
3
a_{3}
a
3
,
⋯
\cdots
⋯
become periodic after a finite number of terms?
1
1
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N 1
Show that the sequence
{
a
n
}
n
≥
1
\{a_{n}\}_{n \ge 1}
{
a
n
}
n
≥
1
defined by
a
n
=
⌊
n
2
⌋
a_{n}=\lfloor n\sqrt{2}\rfloor
a
n
=
⌊
n
2
⌋
contains an infinite number of integer powers of
2
2
2
.