MathDB
Problems
Contests
National and Regional Contests
PEN Problems
PEN I Problems
PEN I Problems
Part of
PEN Problems
Subcontests
(19)
20
1
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I 20
Find all integer solutions of the equation
⌊
x
1
!
⌋
+
⌊
x
2
!
⌋
+
⋯
+
⌊
x
10
!
⌋
=
1001.
\left\lfloor \frac{x}{1!}\right\rfloor+\left\lfloor \frac{x}{2!}\right\rfloor+\cdots+\left\lfloor \frac{x}{10!}\right\rfloor =1001.
⌊
1
!
x
⌋
+
⌊
2
!
x
⌋
+
⋯
+
⌊
10
!
x
⌋
=
1001.
19
1
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I 19
Let
a
,
b
,
c
a, b, c
a
,
b
,
c
, and
d
d
d
be real numbers. Suppose that
⌊
n
a
⌋
+
⌊
n
b
⌋
=
⌊
n
c
⌋
+
⌊
n
d
⌋
\lfloor na\rfloor +\lfloor nb\rfloor =\lfloor nc\rfloor +\lfloor nd\rfloor
⌊
na
⌋
+
⌊
nb
⌋
=
⌊
n
c
⌋
+
⌊
n
d
⌋
for all positive integers
n
n
n
. Show that at least one of
a
+
b
a+b
a
+
b
,
a
−
c
a-c
a
−
c
,
a
−
d
a-d
a
−
d
is an integer.
18
1
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I 18
Do there exist irrational numbers
a
,
b
>
1
a, b>1
a
,
b
>
1
and
⌊
a
m
⌋
≠
⌊
b
n
⌋
\lfloor a^{m}\rfloor \not=\lfloor b^{n}\rfloor
⌊
a
m
⌋
=
⌊
b
n
⌋
for any positive integers
m
m
m
and
n
n
n
?
17
1
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I 17
Determine all real numbers
a
a
a
such that
4
⌊
a
n
⌋
=
n
+
⌊
a
⌊
a
n
⌋
⌋
for all
n
∈
N
.
4\lfloor an\rfloor =n+\lfloor a\lfloor an\rfloor \rfloor \; \text{for all}\; n \in \mathbb{N}.
4
⌊
an
⌋
=
n
+
⌊
a
⌊
an
⌋⌋
for all
n
∈
N
.
16
1
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I 16
Prove or disprove that there exists a positive real number
u
u
u
such that
⌊
u
n
⌋
−
n
\lfloor u^n \rfloor -n
⌊
u
n
⌋
−
n
is an even integer for all positive integer
n
n
n
.
15
1
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I 15
Find the total number of different integer values the function
f
(
x
)
=
⌊
x
⌋
+
⌊
2
x
⌋
+
⌊
5
x
3
⌋
+
⌊
3
x
⌋
+
⌊
4
x
⌋
f(x) = \lfloor x\rfloor+\lfloor 2x\rfloor+\left\lfloor \frac{5x}{3}\right\rfloor+\lfloor 3x\rfloor+\lfloor 4x\rfloor
f
(
x
)
=
⌊
x
⌋
+
⌊
2
x
⌋
+
⌊
3
5
x
⌋
+
⌊
3
x
⌋
+
⌊
4
x
⌋
takes for real numbers
x
x
x
with
0
≤
x
≤
100
0 \leq x \leq 100
0
≤
x
≤
100
.
14
1
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I 14
Let
a
,
b
,
n
a, b, n
a
,
b
,
n
be positive integers with
gcd
(
a
,
b
)
=
1
\gcd(a, b)=1
g
cd
(
a
,
b
)
=
1
. Prove that
∑
k
{
a
k
+
b
n
}
=
n
−
1
2
,
\sum_{k}\left\{ \frac{ak+b}{n}\right\}=\frac{n-1}{2},
k
∑
{
n
ak
+
b
}
=
2
n
−
1
,
where
k
k
k
runs through a complete system of residues modulo
m
m
m
.
13
1
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I 13
Suppose that
n
≥
2
n \ge 2
n
≥
2
. Prove that
∑
k
=
2
n
⌊
n
2
k
⌋
=
∑
k
=
n
+
1
n
2
⌊
n
2
k
⌋
.
\sum_{k=2}^{n}\left\lfloor \frac{n^{2}}{k}\right\rfloor = \sum_{k=n+1}^{n^{2}}\left\lfloor \frac{n^{2}}{k}\right\rfloor.
k
=
2
∑
n
⌊
k
n
2
⌋
=
k
=
n
+
1
∑
n
2
⌊
k
n
2
⌋
.
12
1
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I 12
Let
p
=
4
k
+
1
p=4k+1
p
=
4
k
+
1
be a prime. Show that
∑
i
=
1
k
⌊
i
p
⌋
=
p
2
−
1
12
.
\sum^{k}_{i=1}\left \lfloor \sqrt{ ip }\right \rfloor = \frac{p^{2}-1}{12}.
i
=
1
∑
k
⌊
i
p
⌋
=
12
p
2
−
1
.
11
1
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I 11
Let
p
p
p
be a prime number of the form
4
k
+
1
4k+1
4
k
+
1
. Show that
∑
i
=
1
p
−
1
(
⌊
2
i
2
p
⌋
−
2
⌊
i
2
p
⌋
)
=
p
−
1
2
.
\sum^{p-1}_{i=1}\left( \left \lfloor \frac{2i^{2}}{p}\right \rfloor-2\left \lfloor \frac{i^{2}}{p}\right \rfloor \right) = \frac{p-1}{2}.
i
=
1
∑
p
−
1
(
⌊
p
2
i
2
⌋
−
2
⌊
p
i
2
⌋
)
=
2
p
−
1
.
10
1
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I 10
Show that for all primes
p
p
p
,
∑
k
=
1
p
−
1
⌊
k
3
p
⌋
=
(
p
+
1
)
(
p
−
1
)
(
p
−
2
)
4
.
\sum^{p-1}_{k=1}\left \lfloor \frac{k^{3}}{p}\right \rfloor =\frac{(p+1)(p-1)(p-2)}{4}.
k
=
1
∑
p
−
1
⌊
p
k
3
⌋
=
4
(
p
+
1
)
(
p
−
1
)
(
p
−
2
)
.
9
1
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I 9
Show that for all positive integers
m
m
m
and
n
n
n
,
gcd
(
m
,
n
)
=
m
+
n
−
m
n
+
2
∑
k
=
0
m
−
1
⌊
k
n
m
⌋
.
\gcd(m, n) = m+n-mn+2\sum^{m-1}_{k=0}\left \lfloor \frac{kn}{m}\right \rfloor.
g
cd
(
m
,
n
)
=
m
+
n
−
mn
+
2
k
=
0
∑
m
−
1
⌊
m
kn
⌋
.
8
1
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I 8
Prove that
⌊
n
3
+
n
+
1
3
+
n
+
2
3
⌋
=
⌊
27
n
+
26
3
⌋
\lfloor \sqrt[3]{n}+\sqrt[3]{n+1}+\sqrt[3]{n+2}\rfloor =\lfloor \sqrt[3]{27n+26}\rfloor
⌊
3
n
+
3
n
+
1
+
3
n
+
2
⌋
=
⌊
3
27
n
+
26
⌋
for all positive integers
n
n
n
.
7
1
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I 7
Prove that for all positive integers
n
n
n
,
⌊
n
3
+
n
+
1
3
⌋
=
⌊
8
n
+
3
3
⌋
.
\lfloor \sqrt[3]{n}+\sqrt[3]{n+1}\rfloor =\lfloor \sqrt[3]{8n+3}\rfloor.
⌊
3
n
+
3
n
+
1
⌋
=
⌊
3
8
n
+
3
⌋
.
6
1
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I 6
Prove that for all positive integers
n
n
n
,
⌊
n
+
n
+
1
+
n
+
2
⌋
=
⌊
9
n
+
8
⌋
.
\lfloor \sqrt{n}+\sqrt{n+1}+\sqrt{n+2}\rfloor =\lfloor \sqrt{9n+8}\rfloor.
⌊
n
+
n
+
1
+
n
+
2
⌋
=
⌊
9
n
+
8
⌋
.
5
1
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I 5
Find all real numbers
α
\alpha
α
for which the equality
⌊
n
+
n
+
α
⌋
=
⌊
4
n
+
1
⌋
\lfloor \sqrt{n}+\sqrt{n+\alpha}\rfloor =\lfloor \sqrt{4n+1}\rfloor
⌊
n
+
n
+
α
⌋
=
⌊
4
n
+
1
⌋
holds for all positive integers
n
n
n
.
4
1
Hide problems
I 4
Show that for all positive integers
n
n
n
,
⌊
n
+
n
+
1
⌋
=
⌊
4
n
+
1
⌋
=
⌊
4
n
+
2
⌋
=
⌊
4
n
+
3
⌋
.
\lfloor \sqrt{n}+\sqrt{n+1}\rfloor =\lfloor \sqrt{4n+1}\rfloor =\lfloor \sqrt{4n+2}\rfloor =\lfloor \sqrt{4n+3}\rfloor.
⌊
n
+
n
+
1
⌋
=
⌊
4
n
+
1
⌋
=
⌊
4
n
+
2
⌋
=
⌊
4
n
+
3
⌋
.
3
1
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I 3
Prove that for any positive integer
n
n
n
,
⌊
n
+
1
2
⌋
+
⌊
n
+
2
4
⌋
+
⌊
n
+
4
8
⌋
+
⌊
n
+
8
16
⌋
+
⋯
=
n
.
\left\lfloor \frac{n+1}{2}\right\rfloor+\left\lfloor \frac{n+2}{4}\right\rfloor+\left\lfloor \frac{n+4}{8}\right\rfloor+\left\lfloor \frac{n+8}{16}\right\rfloor+\cdots = n.
⌊
2
n
+
1
⌋
+
⌊
4
n
+
2
⌋
+
⌊
8
n
+
4
⌋
+
⌊
16
n
+
8
⌋
+
⋯
=
n
.
2
1
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I 2
Prove that for any positive integer
n
n
n
,
⌊
n
3
⌋
+
⌊
n
+
2
6
⌋
+
⌊
n
+
4
6
⌋
=
⌊
n
2
⌋
+
⌊
n
+
3
6
⌋
.
\left\lfloor \frac{n}{3}\right\rfloor+\left\lfloor \frac{n+2}{6}\right\rfloor+\left\lfloor \frac{n+4}{6}\right\rfloor = \left\lfloor \frac{n}{2}\right\rfloor+\left\lfloor \frac{n+3}{6}\right\rfloor .
⌊
3
n
⌋
+
⌊
6
n
+
2
⌋
+
⌊
6
n
+
4
⌋
=
⌊
2
n
⌋
+
⌊
6
n
+
3
⌋
.