MathDB
Problems
Contests
National and Regional Contests
Netherlands Contests
Dutch Mathematical Olympiad
1972 Dutch Mathematical Olympiad
1972 Dutch Mathematical Olympiad
Part of
Dutch Mathematical Olympiad
Subcontests
(5)
4
1
Hide problems
set of k points from n on a circle
On a circle with radius
1
1
1
the points
A
1
,
A
2
,
.
.
.
,
A
n
A_1, A_2,..., A_n
A
1
,
A
2
,
...
,
A
n
lie such that every arc
A
i
A
i
+
i
A_iA_{i+i}
A
i
A
i
+
i
has length
2
π
n
=
a
\frac{2\pi}{n}= a
n
2
π
=
a
. Given that there exists a set
V
V
V
consisting of
k
k
k
of these points (
k
<
n
k < n
k
<
n
), which has the property that each of the arc lengths
a
a
a
,
2
a
2a
2
a
,
.
.
.
,...
,
...
,
(
n
−
1
)
a
(n- 1)a
(
n
−
1
)
a
can be obtained in exactly one way be taken as the length of an arc traversed in a positive sense, beginning and ending in a point of
V
V
V
. Express
n
n
n
in terms of
k
k
k
and give the set
V
V
V
for the case
n
=
7
n = 7
n
=
7
.
2
1
Hide problems
x <= y => f(x) <= f(y) , f(f(x)) = x
Prove that there exists exactly one function
ƒ
ƒ
ƒ
which is defined for all
x
∈
R
x \in R
x
∈
R
, and for which holds:
∙
\bullet
∙
x
≤
y
⇒
f
(
x
)
≤
f
(
y
)
x \le y \Rightarrow f(x) \le f(y)
x
≤
y
⇒
f
(
x
)
≤
f
(
y
)
, for all
x
,
y
∈
R
x, y \in R
x
,
y
∈
R
, and
∙
\bullet
∙
f
(
f
(
x
)
)
=
x
f(f(x)) = x
f
(
f
(
x
))
=
x
, for all
x
∈
R
x \in R
x
∈
R
.
3
1
Hide problems
compare volumes , regular tetrahedra
A
B
C
D
ABCD
A
BC
D
is a regular tetrahedron. The points
P
,
Q
,
R
P,Q,R
P
,
Q
,
R
and
S
S
S
lie outside this tetrahedron in such a way that
A
B
C
P
ABCP
A
BCP
,
A
B
D
Q
ABDQ
A
B
D
Q
,
A
C
D
R
ACDR
A
C
D
R
and
B
C
D
S
BCDS
BC
D
S
are regular tetrahedra. Prove that the volume of the tetrahedron
P
Q
R
S
PQRS
PQRS
is less than the sum of the volumes of
A
B
C
P
ABCP
A
BCP
,
A
B
D
Q
ABDQ
A
B
D
Q
,
A
C
D
R
ACDR
A
C
D
R
,
B
C
D
S
BCDS
BC
D
S
and
A
B
C
D
ABCD
A
BC
D
.
1
1
Hide problems
any equilateral triangle can be divided into n>6 equilaterals
Prove that for every
n
∈
N
n \in N
n
∈
N
,
n
>
6
n > 6
n
>
6
, every equilateral triangle can be divided into
n
n
n
pieces, which are also equilateral triangles.
5
1
Hide problems
AP/BP in terms of angles of ABC
Given is an acute-angled triangle
A
B
C
ABC
A
BC
with angles
α
\alpha
α
,
β
\beta
β
and
γ
\gamma
γ
. On side
A
B
AB
A
B
lies a point
P
P
P
such that the line connecting the feet of the perpendiculars from
P
P
P
on
A
C
AC
A
C
and
B
C
BC
BC
is parallel to
A
B
AB
A
B
. Express the ratio
A
P
B
P
\frac{AP}{BP}
BP
A
P
in terms of
α
\alpha
α
and
β
\beta
β
.