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Problems
Contests
National and Regional Contests
PEN Problems
PEN F Problems
PEN F Problems
Part of
PEN Problems
Subcontests
(15)
16
1
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F 16
Prove that for any distinct rational numbers
a
,
b
,
c
a, b, c
a
,
b
,
c
, the number
1
(
b
−
c
)
2
+
1
(
c
−
a
)
2
+
1
(
a
−
b
)
2
\frac{1}{(b-c)^{2}}+\frac{1}{(c-a)^{2}}+\frac{1}{(a-b)^{2}}
(
b
−
c
)
2
1
+
(
c
−
a
)
2
1
+
(
a
−
b
)
2
1
is the square of some rational number.
15
1
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F 15
Find all rational numbers
k
k
k
such that
0
≤
k
≤
1
2
0 \le k \le \frac{1}{2}
0
≤
k
≤
2
1
and
cos
k
π
\cos k \pi
cos
kπ
is rational.
14
1
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F 14
Let
k
k
k
and
m
m
m
be positive integers. Show that
S
(
m
,
k
)
=
∑
n
=
1
∞
1
n
(
m
n
+
k
)
S(m, k)=\sum_{n=1}^{\infty}\frac{1}{n(mn+k)}
S
(
m
,
k
)
=
n
=
1
∑
∞
n
(
mn
+
k
)
1
is rational if and only if
m
m
m
divides
k
k
k
.
13
1
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F 13
Prove that numbers of the form
a
1
1
!
+
a
2
2
!
+
a
3
3
!
+
⋯
,
\frac{a_{1}}{1!}+\frac{a_{2}}{2!}+\frac{a_{3}}{3!}+\cdots,
1
!
a
1
+
2
!
a
2
+
3
!
a
3
+
⋯
,
where
0
≤
a
i
≤
i
−
1
(
i
=
2
,
3
,
4
,
⋯
)
0 \le a_{i}\le i-1 \;(i=2, 3, 4, \cdots)
0
≤
a
i
≤
i
−
1
(
i
=
2
,
3
,
4
,
⋯
)
are rational if and only if starting from some
i
i
i
on all the
a
i
a_{i}
a
i
's are either equal to
0
0
0
( in which case the sum is finite) or all are equal to
i
−
1
i-1
i
−
1
.
12
1
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F 12
Does there exist a circle and an infinite set of points on it such that the distance between any two points of the set is rational?
11
1
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F 11
Let
S
=
{
x
0
,
x
1
,
⋯
,
x
n
}
⊂
[
0
,
1
]
S=\{x_0, x_1, \cdots, x_n\} \subset [0,1]
S
=
{
x
0
,
x
1
,
⋯
,
x
n
}
⊂
[
0
,
1
]
be a finite set of real numbers with
x
0
=
0
x_{0}=0
x
0
=
0
and
x
1
=
1
x_{1}=1
x
1
=
1
, such that every distance between pairs of elements occurs at least twice, except for the distance
1
1
1
. Prove that all of the
x
i
x_i
x
i
are rational.
10
1
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F 10
The set
S
S
S
is a finite subset of
[
0
,
1
]
[0,1]
[
0
,
1
]
with the following property: for all
s
∈
S
s\in S
s
∈
S
, there exist
a
,
b
∈
S
∪
{
0
,
1
}
a,b\in S\cup\{0,1\}
a
,
b
∈
S
∪
{
0
,
1
}
with
a
,
b
≠
s
a, b\neq s
a
,
b
=
s
such that s \equal{}\frac{a\plus{}b}{2}. Prove that all the numbers in
S
S
S
are rational.
9
1
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F 9
Prove that every positive rational number can be represented in the form
a
3
+
b
3
c
3
+
d
3
\frac{a^{3}+b^{3}}{c^{3}+d^{3}}
c
3
+
d
3
a
3
+
b
3
for some positive integers
a
,
b
,
c
a, b, c
a
,
b
,
c
, and
d
d
d
.
8
1
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F 8
Find all polynomials
W
W
W
with real coefficients possessing the following property: if
x
+
y
x+y
x
+
y
is a rational number, then
W
(
x
)
+
W
(
y
)
W(x)+W(y)
W
(
x
)
+
W
(
y
)
is rational.
7
1
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F 7
If
x
x
x
is a positive rational number, show that
x
x
x
can be uniquely expressed in the form
x
=
a
1
+
a
2
2
!
+
a
3
3
!
+
⋯
,
x=a_{1}+\frac{a_{2}}{2!}+\frac{a_{3}}{3!}+\cdots,
x
=
a
1
+
2
!
a
2
+
3
!
a
3
+
⋯
,
where
a
1
a
2
,
⋯
a_{1}a_{2},\cdots
a
1
a
2
,
⋯
are integers,
0
≤
a
n
≤
n
−
1
0 \le a_{n}\le n-1
0
≤
a
n
≤
n
−
1
for
n
>
1
n>1
n
>
1
, and the series terminates. Show also that
x
x
x
can be expressed as the sum of reciprocals of different integers, each of which is greater than
1
0
6
10^{6}
1
0
6
.
6
1
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F 6
Let
x
,
y
,
z
x, y, z
x
,
y
,
z
non-zero real numbers such that
x
y
xy
x
y
,
y
z
yz
yz
,
z
x
zx
z
x
are rational. [*] Show that the number
x
2
+
y
2
+
z
2
x^{2}+y^{2}+z^{2}
x
2
+
y
2
+
z
2
is rational. [*] If the number
x
3
+
y
3
+
z
3
x^{3}+y^{3}+z^{3}
x
3
+
y
3
+
z
3
is also rational, show that
x
x
x
,
y
y
y
,
z
z
z
are rational.
5
1
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F 5
Prove that there is no positive rational number
x
x
x
such that
x
⌊
x
⌋
=
9
2
.
x^{\lfloor x\rfloor }=\frac{9}{2}.
x
⌊
x
⌋
=
2
9
.
4
1
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F 4
Suppose that
tan
α
=
p
q
\tan \alpha =\frac{p}{q}
tan
α
=
q
p
, where
p
p
p
and
q
q
q
are integers and
q
≠
0
q \neq 0
q
=
0
. Prove the number
tan
β
\tan \beta
tan
β
for which
tan
2
β
=
tan
3
α
\tan 2\beta =\tan 3\alpha
tan
2
β
=
tan
3
α
is rational only when
p
2
+
q
2
p^2 +q^2
p
2
+
q
2
is the square of an integer.
2
1
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F 2
Find all
x
x
x
and
y
y
y
which are rational multiples of
π
\pi
π
with
0
<
x
<
y
<
π
2
0<x<y<\frac{\pi}{2}
0
<
x
<
y
<
2
π
and
tan
x
+
tan
y
=
2
\tan x+\tan y =2
tan
x
+
tan
y
=
2
.
1
1
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F 1
Suppose that a rectangle with sides
a
a
a
and
b
b
b
is arbitrarily cut into
n
n
n
squares with sides
x
1
,
…
,
x
n
x_{1},\ldots,x_{n}
x
1
,
…
,
x
n
. Show that
x
i
a
∈
Q
\frac{x_{i}}{a}\in\mathbb{Q}
a
x
i
∈
Q
and
x
i
b
∈
Q
\frac{x_{i}}{b}\in\mathbb{Q}
b
x
i
∈
Q
for all
i
∈
{
1
,
⋯
,
n
}
i\in\{1,\cdots, n\}
i
∈
{
1
,
⋯
,
n
}
.