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Contests
National and Regional Contests
PEN Problems
PEN S Problems
PEN S Problems
Part of
PEN Problems
Subcontests
(37)
38
1
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S 38
The function
μ
:
N
→
C
\mu: \mathbb{N}\to \mathbb{C}
μ
:
N
→
C
is defined by
μ
(
n
)
=
∑
k
∈
R
n
(
cos
2
k
π
n
+
i
sin
2
k
π
n
)
,
\mu(n) = \sum^{}_{k \in R_{n}}\left( \cos \frac{2k\pi}{n}+i \sin \frac{2k\pi}{n}\right),
μ
(
n
)
=
k
∈
R
n
∑
(
cos
n
2
kπ
+
i
sin
n
2
kπ
)
,
where
R
n
=
{
k
∈
N
∣
1
≤
k
≤
n
,
gcd
(
k
,
n
)
=
1
}
R_{n}=\{ k \in \mathbb{N}\vert 1 \le k \le n, \gcd(k, n)=1 \}
R
n
=
{
k
∈
N
∣1
≤
k
≤
n
,
g
cd
(
k
,
n
)
=
1
}
. Show that
μ
(
n
)
\mu(n)
μ
(
n
)
is an integer for all positive integer
n
n
n
.
37
1
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S 37
Let
n
n
n
and
k
k
k
are integers with
n
>
0
n>0
n
>
0
. Prove that
−
1
2
n
∑
m
=
1
n
−
1
cot
π
m
n
sin
2
π
k
m
n
=
{
k
n
−
⌊
k
n
⌋
−
1
2
if
k
∣
n
0
otherwise
.
-\frac{1}{2n}\sum^{n-1}_{m=1}\cot \frac{\pi m}{n}\sin \frac{2\pi km}{n}= \begin{cases}\tfrac{k}{n}-\lfloor\tfrac{k}{n}\rfloor-\frac12 & \text{if }k|n \\ 0 & \text{otherwise}\end{cases}.
−
2
n
1
m
=
1
∑
n
−
1
cot
n
πm
sin
n
2
πkm
=
{
n
k
−
⌊
n
k
⌋
−
2
1
0
if
k
∣
n
otherwise
.
36
1
Hide problems
S 36
For every natural number
n
n
n
, denote
Q
(
n
)
Q(n)
Q
(
n
)
the sum of the digits in the decimal representation of
n
n
n
. Prove that there are infinitely many natural numbers
k
k
k
with
Q
(
3
k
)
>
Q
(
3
k
+
1
)
Q(3^{k})>Q(3^{k+1})
Q
(
3
k
)
>
Q
(
3
k
+
1
)
.
35
1
Hide problems
S 35
Counting from the right end, what is the
2500
2500
2500
th digit of
10000
!
10000!
10000
!
?
34
1
Hide problems
S 34
Let
S
n
S_{n}
S
n
be the sum of the digits of
2
n
2^n
2
n
. Prove or disprove that
S
n
+
1
=
S
n
S_{n+1}=S_{n}
S
n
+
1
=
S
n
for some positive integer
n
n
n
.
33
1
Hide problems
S 33
Four consecutive even numbers are removed from the set
A
=
{
1
,
2
,
3
,
⋯
,
n
}
.
A=\{ 1, 2, 3, \cdots, n \}.
A
=
{
1
,
2
,
3
,
⋯
,
n
}
.
If the arithmetic mean of the remaining numbers is
51.5625
51.5625
51.5625
, which four numbers were removed?
32
1
Hide problems
S 32
Alice and Bob play the following number-guessing game. Alice writes down a list of positive integers
x
1
x_{1}
x
1
,
⋯
\cdots
⋯
,
x
n
x_{n}
x
n
, but does not reveal them to Bob, who will try to determine the numbers by asking Alice questions. Bob chooses a list of positive integers
a
1
a_{1}
a
1
,
⋯
\cdots
⋯
,
a
n
a_{n}
a
n
and asks Alice to tell him the value of
a
1
x
1
+
⋯
+
a
n
x
n
a_{1}x_{1}+\cdots+a_{n}x_{n}
a
1
x
1
+
⋯
+
a
n
x
n
. Then Bob chooses another list of positive integers
b
1
b_{1}
b
1
,
⋯
\cdots
⋯
,
b
n
b_{n}
b
n
and asks Alice for
b
1
x
1
+
⋯
+
b
n
x
n
b_{1}x_{1}+\cdots+b_{n}x_{n}
b
1
x
1
+
⋯
+
b
n
x
n
. Play continues in this way until Bob is able to determine Alice's numbers. How many rounds will Bob need in order to determine Alice's numbers?
31
1
Hide problems
S 31
Is there a
3
×
3
3 \times 3
3
×
3
magic square consisting of distinct Fibonacci numbers (both
f
1
f_{1}
f
1
and
f
2
f_{2}
f
2
may be used; thus two
1
1
1
s are allowed)?
30
1
Hide problems
S 30
For how many positive integers
n
n
n
is
(
1999
+
1
2
)
n
+
(
2000
+
1
2
)
n
\left( 1999+\frac{1}{2}\right)^{n}+\left(2000+\frac{1}{2}\right)^{n}
(
1999
+
2
1
)
n
+
(
2000
+
2
1
)
n
an integer?
29
1
Hide problems
S 29
What is the rightmost nonzero digit of
1000000
!
1000000!
1000000
!
?
28
1
Hide problems
S 28
Let
A
A
A
be the set of the
16
16
16
first positive integers. Find the least positive integer
k
k
k
satisfying the condition: In every
k
k
k
-subset of
A
A
A
, there exist two distinct
a
,
b
∈
A
a, b \in A
a
,
b
∈
A
such that
a
2
+
b
2
a^2 + b^2
a
2
+
b
2
is prime.
27
1
Hide problems
S 27
Which integers have the following property? If the final digit is deleted, the integer is divisible by the new number.
26
1
Hide problems
S 26
Prove that there does not exist a natural number which, upon transfer of its initial digit to the end, is increased five, six or eight times.
24
1
Hide problems
S 24
A number
n
n
n
is called a Niven number, named for Ivan Niven, if it is divisible by the sum of its digits. For example,
24
24
24
is a Niven number. Show that it is not possible to have more than
20
20
20
consecutive Niven numbers.
23
1
Hide problems
S 23
Observe that
1
1
+
1
3
=
4
3
,
4
2
+
3
2
=
5
2
,
\frac{1}{1}+\frac{1}{3}=\frac{4}{3}, \;\; 4^{2}+3^{2}=5^{2},
1
1
+
3
1
=
3
4
,
4
2
+
3
2
=
5
2
,
1
3
+
1
5
=
8
15
,
8
2
+
15
2
=
17
2
,
\frac{1}{3}+\frac{1}{5}=\frac{8}{15}, \;\; 8^{2}+{15}^{2}={17}^{2},
3
1
+
5
1
=
15
8
,
8
2
+
15
2
=
17
2
,
1
5
+
1
7
=
12
35
,
12
2
+
35
2
=
37
2
.
\frac{1}{5}+\frac{1}{7}=\frac{12}{35}, \;\;{12}^{2}+{35}^{2}={37}^{2}.
5
1
+
7
1
=
35
12
,
12
2
+
35
2
=
37
2
.
State and prove a generalization suggested by these examples.
22
1
Hide problems
S 22
The decimal expression of the natural number
a
a
a
consists of
n
n
n
digits, while that of
a
3
a^3
a
3
consists of
m
m
m
digits. Can
n
+
m
n+m
n
+
m
be equal to
2001
2001
2001
?
21
1
Hide problems
S 21
Find, with proof, the number of positive integers whose base-
n
n
n
representation consists of distinct digits with the property that, except for the leftmost digit, every digit differs by
±
1
\pm 1
±
1
from some digit further to the left.
20
1
Hide problems
S 20
Let
n
n
n
be a positive integer that is not a perfect cube. Define real numbers
a
a
a
,
b
b
b
,
c
c
c
by
a
=
n
3
,
b
=
1
a
−
⌊
a
⌋
,
c
=
1
b
−
⌊
b
⌋
.
a=\sqrt[3]{n}, \; b=\frac{1}{a-\lfloor a\rfloor}, \; c=\frac{1}{b-\lfloor b\rfloor}.
a
=
3
n
,
b
=
a
−
⌊
a
⌋
1
,
c
=
b
−
⌊
b
⌋
1
.
Prove that there are infinitely many such integers
n
n
n
with the property that there exist integers
r
r
r
,
s
s
s
,
t
t
t
, not all zero, such that
r
a
+
s
b
+
t
c
=
0
ra+sb+tc=0
r
a
+
s
b
+
t
c
=
0
.
19
1
Hide problems
S 19
Determine all pairs
(
a
,
b
)
(a, b)
(
a
,
b
)
of real numbers such that
a
⌊
b
n
⌋
=
b
⌊
a
n
⌋
a\lfloor bn\rfloor =b\lfloor an\rfloor
a
⌊
bn
⌋
=
b
⌊
an
⌋
for all positive integer
n
n
n
.
18
1
Hide problems
S 18
Denote by
S
S
S
the set of all primes
p
p
p
such that the decimal representation of
1
p
\frac{1}{p}
p
1
has the fundamental period of divisible by
3
3
3
. For every
p
∈
S
p \in S
p
∈
S
such that
1
p
\frac{1}{p}
p
1
has the fundamental period
3
r
3r
3
r
one may write
1
p
=
0.
a
1
a
2
⋯
a
3
r
a
1
a
2
⋯
a
3
r
⋯
,
\frac{1}{p}= 0.a_{1}a_{2}\cdots a_{3r}a_{1}a_{2}\cdots a_{3r}\cdots,
p
1
=
0.
a
1
a
2
⋯
a
3
r
a
1
a
2
⋯
a
3
r
⋯
,
where
r
=
r
(
p
)
r=r(p)
r
=
r
(
p
)
. For every
p
∈
S
p \in S
p
∈
S
and every integer
k
≥
1
k \ge 1
k
≥
1
define
f
(
k
,
p
)
=
a
k
+
a
k
+
r
(
p
)
+
a
k
+
2
r
(
p
)
.
f(k, p)=a_{k}+a_{k+r(p)}+a_{k+2r(p)}.
f
(
k
,
p
)
=
a
k
+
a
k
+
r
(
p
)
+
a
k
+
2
r
(
p
)
.
[*] Prove that
S
S
S
is finite. [*] Find the highest value of
f
(
k
,
p
)
f(k, p)
f
(
k
,
p
)
for
k
≥
1
k \ge 1
k
≥
1
and
p
∈
S
p \in S
p
∈
S
.
17
1
Hide problems
S 17
Determine the maximum value of
m
2
+
n
2
m^{2}+n^{2}
m
2
+
n
2
, where
m
m
m
and
n
n
n
are integers satisfying
m
,
n
∈
{
1
,
2
,
.
.
.
,
1981
}
m,n\in \{1,2,...,1981\}
m
,
n
∈
{
1
,
2
,
...
,
1981
}
and
(
n
2
−
m
n
−
m
2
)
2
=
1.
(n^{2}-mn-m^{2})^{2}=1.
(
n
2
−
mn
−
m
2
)
2
=
1.
16
1
Hide problems
S 16
Show that if
a
a
a
and
b
b
b
are positive integers, then
(
a
+
1
2
)
n
+
(
b
+
1
2
)
n
\left( a+\frac{1}{2}\right)^{n}+\left( b+\frac{1}{2}\right)^{n}
(
a
+
2
1
)
n
+
(
b
+
2
1
)
n
is an integer for only finitely many positive integer
n
n
n
.
15
1
Hide problems
S 15
Let
α
(
n
)
\alpha(n)
α
(
n
)
be the number of digits equal to one in the dyadic representation of a positive integer
n
n
n
. Prove that [*] the inequality
α
(
n
2
)
≤
1
2
α
(
n
)
(
1
+
α
(
n
)
)
\alpha(n^2 ) \le \frac{1}{2} \alpha(n) (1+\alpha(n))
α
(
n
2
)
≤
2
1
α
(
n
)
(
1
+
α
(
n
))
holds, [*] equality is attained for infinitely
n
∈
N
n\in\mathbb{N}
n
∈
N
, [*] there exists a sequence
{
n
i
}
\{n_i\}
{
n
i
}
such that
lim
i
→
∞
α
(
n
i
2
)
α
(
n
i
)
=
0
\lim_{i \to \infty} \frac{ \alpha({n_{i}}^2 )}{ \alpha(n_{i}) } = 0
lim
i
→
∞
α
(
n
i
)
α
(
n
i
2
)
=
0
.
14
1
Hide problems
S 14
Let
p
p
p
be an odd prime. Determine positive integers
x
x
x
and
y
y
y
for which
x
≤
y
x \le y
x
≤
y
and
2
p
−
x
−
y
\sqrt{2p}-\sqrt{x}-\sqrt{y}
2
p
−
x
−
y
is nonnegative and as small as possible.
13
1
Hide problems
S 13
The sum of the digits of a natural number
n
n
n
is denoted by
S
(
n
)
S(n)
S
(
n
)
. Prove that
S
(
8
n
)
≥
1
8
S
(
n
)
S(8n) \ge \frac{1}{8} S(n)
S
(
8
n
)
≥
8
1
S
(
n
)
for each
n
n
n
.
12
1
Hide problems
S 12
Let
a
1
,
1
a
1
,
2
a
1
,
3
…
a
2
,
1
a
2
,
2
a
2
,
3
…
a
3
,
1
a
3
,
2
a
3
,
3
…
⋮
⋮
⋮
⋱
\begin{array}{cccc}a_{1,1}& a_{1,2}& a_{1,3}& \dots \\ a_{2,1}& a_{2,2}& a_{2,3}& \dots \\ a_{3,1}& a_{3,2}& a_{3,3}& \dots \\ \vdots & \vdots & \vdots & \ddots \end{array}
a
1
,
1
a
2
,
1
a
3
,
1
⋮
a
1
,
2
a
2
,
2
a
3
,
2
⋮
a
1
,
3
a
2
,
3
a
3
,
3
⋮
…
…
…
⋱
be a doubly infinite array of positive integers, and suppose each positive integer appears exactly eight times in the array. Prove that
a
m
,
n
>
m
n
a_{m,n}> mn
a
m
,
n
>
mn
for some pair of positive integers
(
m
,
n
)
(m,n)
(
m
,
n
)
.
11
1
Hide problems
S 11
For each positive integer
n
n
n
, prove that there are two consecutive positive integers each of which is the product of
n
n
n
positive integers greater than
1
1
1
.
10
1
Hide problems
S 10
Let
p
p
p
be an odd prime. Show that there is at most one non-degenerate integer triangle with perimeter
4
p
4p
4
p
and integer area. Characterize those primes for which such triangle exist.
9
1
Hide problems
S 9
Suppose that
∏
n
=
1
1996
(
1
+
n
x
3
n
)
=
1
+
a
1
x
k
1
+
a
2
x
k
2
+
⋯
+
a
m
x
k
m
\prod_{n=1}^{1996}(1+nx^{3^{n}}) = 1+a_{1}x^{k_{1}}+a_{2}x^{k_{2}}+\cdots+a_{m}x^{k_{m}}
n
=
1
∏
1996
(
1
+
n
x
3
n
)
=
1
+
a
1
x
k
1
+
a
2
x
k
2
+
⋯
+
a
m
x
k
m
where
a
1
a_{1}
a
1
,
a
2
a_{2}
a
2
,...,
a
m
a_{m}
a
m
are nonzero and
k
1
<
k
2
<
⋯
<
k
m
k_{1}< k_{2}< \cdots < k_{m}
k
1
<
k
2
<
⋯
<
k
m
. Find
a
1996
a_{1996}
a
1996
.
8
1
Hide problems
S 8
The set
S
=
{
1
n
∣
n
∈
N
}
S=\{ \frac{1}{n} \; \vert \; n \in \mathbb{N} \}
S
=
{
n
1
∣
n
∈
N
}
contains arithmetic progressions of various lengths. For instance,
1
20
\frac{1}{20}
20
1
,
1
8
\frac{1}{8}
8
1
,
1
5
\frac{1}{5}
5
1
is such a progression of length
3
3
3
and common difference
3
40
\frac{3}{40}
40
3
. Moreover, this is a maximal progression in
S
S
S
since it cannot be extended to the left or the right within
S
S
S
(
11
40
\frac{11}{40}
40
11
and
−
1
40
\frac{-1}{40}
40
−
1
not being members of
S
S
S
). Prove that for all
n
∈
N
n \in \mathbb{N}
n
∈
N
, there exists a maximal arithmetic progression of length
n
n
n
in
S
S
S
.
7
1
Hide problems
S 7
Let
n
n
n
be a positive integer. Show that
∑
k
=
1
n
tan
2
k
π
2
n
+
1
\sum^{n}_{k=1}\tan^{2}\frac{k \pi}{2n+1}
k
=
1
∑
n
tan
2
2
n
+
1
kπ
is an odd integer.
6
1
Hide problems
S 6
Suppose that
x
x
x
and
y
y
y
are complex numbers such that
x
n
−
y
n
x
−
y
\frac{x^{n}-y^{n}}{x-y}
x
−
y
x
n
−
y
n
are integers for some four consecutive positive integers
n
n
n
. Prove that it is an integer for all positive integers
n
n
n
.
5
1
Hide problems
S 5
Suppose that both
x
3
−
x
x^{3}-x
x
3
−
x
and
x
4
−
x
x^{4}-x
x
4
−
x
are integers for some real number
x
x
x
. Show that
x
x
x
is an integer.
4
1
Hide problems
S 4
If
x
x
x
is a real number such that
x
2
−
x
x^2 -x
x
2
−
x
is an integer, and for some
n
≥
3
n \ge 3
n
≥
3
,
x
n
−
x
x^n -x
x
n
−
x
is also an integer, prove that
x
x
x
is an integer.
3
1
Hide problems
S 3
Is there a power of
2
2
2
such that it is possible to rearrange the digits giving another power of
2
2
2
?
2
1
Hide problems
S 2
It is given that
2
333
2^{333}
2
333
is a
101
101
101
-digit number whose first digit is
1
1
1
. How many of the numbers
2
k
2^k
2
k
,
1
≤
k
≤
332
1 \le k \le 332
1
≤
k
≤
332
, have first digit
4
4
4
?
1
1
Hide problems
S 1
a) Two positive integers are chosen. The sum is revealed to logician
A
A
A
, and the sum of squares is revealed to logician
B
B
B
. Both
A
A
A
and
B
B
B
are given this information and the information contained in this sentence. The conversation between
A
A
A
and
B
B
B
goes as follows:
B
B
B
starts B: ` I can't tell what they are.' A: ` I can't tell what they are.' B: ` I can't tell what they are.' A: ` I can't tell what they are.' B: ` I can't tell what they are.' A: ` I can't tell what they are.' B: ` Now I can tell what they are.' What are the two numbers? b) When
B
B
B
first says that he cannot tell what the two numbers are,
A
A
A
receives a large amount of information. But when
A
A
A
first says that he cannot tell what the two numbers are,
B
B
B
already knows that
A
A
A
cannot tell what the two numbers are. What good does it do
B
B
B
to listen to
A
A
A
?