MathDB
Problems
Contests
International Contests
Pan African
2003 Pan African
2003 Pan African
Part of
Pan African
Subcontests
(3)
3
2
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10101, 101010101
Does there exists a base in which the numbers of the form:
10101
,
101010101
,
1010101010101
,
⋯
10101, 101010101, 1010101010101,\cdots
10101
,
101010101
,
1010101010101
,
⋯
are all prime numbers?
Another functional equation
Find all functions
f
:
R
→
R
f: R\to R
f
:
R
→
R
such that:
f
(
x
2
)
−
f
(
y
2
)
=
(
x
+
y
)
(
f
(
x
)
−
f
(
y
)
)
,
x
,
y
∈
R
f(x^2)-f(y^2)=(x+y)(f(x)-f(y)), x,y \in R
f
(
x
2
)
−
f
(
y
2
)
=
(
x
+
y
)
(
f
(
x
)
−
f
(
y
))
,
x
,
y
∈
R
2
2
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Circumference
The circumference of a circle is arbitrarily divided into four arcs. The midpoints of the arcs are connected by segments. Show that two of these segments are perpendicular.
divisible by 21
Find all positive integers
n
n
n
such that
21
21
21
divides
2
2
n
+
2
n
+
1
2^{2^n}+2^n+1
2
2
n
+
2
n
+
1
.
1
2
Hide problems
Function N-->N
Let
N
0
=
{
0
,
1
,
2
⋯
}
N_0=\{0, 1, 2 \cdots \}
N
0
=
{
0
,
1
,
2
⋯
}
. Find all functions:
N
0
→
N
0
N_0 \to N_0
N
0
→
N
0
such that: (1)
f
(
n
)
<
f
(
n
+
1
)
f(n) < f(n+1)
f
(
n
)
<
f
(
n
+
1
)
, all
n
∈
N
0
n \in N_0
n
∈
N
0
; (2)
f
(
2
)
=
2
f(2)=2
f
(
2
)
=
2
; (3)
f
(
m
n
)
=
f
(
m
)
f
(
n
)
f(mn)=f(m)f(n)
f
(
mn
)
=
f
(
m
)
f
(
n
)
, all
m
,
n
∈
N
0
m, n \in N_0
m
,
n
∈
N
0
.
functional equation
Let
N
0
=
{
0
,
1
,
2
⋯
}
\mathbb{N}_0=\{0,1,2 \cdots \}
N
0
=
{
0
,
1
,
2
⋯
}
. Does there exist a function f: \mathbb{N}__0 \to \mathbb{N}_0 such that:
f
2003
(
n
)
=
5
n
,
∀
n
∈
N
0
f^{2003}(n)=5n, \forall n \in \mathbb{N}_0
f
2003
(
n
)
=
5
n
,
∀
n
∈
N
0
where we define:
f
1
(
n
)
=
f
(
n
)
f^1(n)=f(n)
f
1
(
n
)
=
f
(
n
)
and
f
k
+
1
(
n
)
=
f
(
f
k
(
n
)
)
f^{k+1}(n)=f(f^k(n))
f
k
+
1
(
n
)
=
f
(
f
k
(
n
))
,
∀
k
∈
N
0
\forall k \in \mathbb{N}_0
∀
k
∈
N
0
?