MathDB
Problems
Contests
National and Regional Contests
Netherlands Contests
Dutch Mathematical Olympiad
1969 Dutch Mathematical Olympiad
1969 Dutch Mathematical Olympiad
Part of
Dutch Mathematical Olympiad
Subcontests
(5)
3
1
Hide problems
ED = EB if AB = BD = DC, AC = BC, AE = AB
Given a quadrilateral
A
B
C
D
ABCD
A
BC
D
with
A
B
=
B
D
=
D
C
AB = BD = DC
A
B
=
B
D
=
D
C
and
A
C
=
B
C
AC = BC
A
C
=
BC
. On
B
C
BC
BC
lies point
E
E
E
such that
A
E
=
A
B
AE = AB
A
E
=
A
B
. Prove that
E
D
=
E
B
ED = EB
E
D
=
EB
.
4
1
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triangle with minimal perimeter , 2 vertices on legs of an angle
An angle
<
4
5
o
< 45^o
<
4
5
o
is given in the plane of the drawing. Furthermore, the projection
P
1
P_1
P
1
of a point
P
P
P
lying above the plane of the drawing and the distance from
P
P
P
to
P
1
P_1
P
1
are given.
P
1
P_1
P
1
lies within the given angle. On the legs of the given angle, construct points
A
A
A
and
B
B
B
, respectively, such that the triangle
P
A
B
PAB
P
A
B
has a minimal perimeter.
5
1
Hide problems
trigonometric sum and revolutions of a wheel
a) Prove that for
n
=
2
,
3
,
4
,
.
.
.
n = 2,3,4,...
n
=
2
,
3
,
4
,
...
holds:
sin
a
+
sin
2
a
+
.
.
.
+
sin
(
n
−
1
)
a
=
cos
a
(
a
2
)
−
cos
(
n
−
1
2
)
a
2
sin
(
a
2
)
\sin a + \sin 2a + ...+ \sin (n-1)a=\frac{\cos a \left(\frac{a}{2}\right) - \cos \left(n-\frac{1}{2}\right) a}{2 \sin \left(\frac{a}{2}\right)}
sin
a
+
sin
2
a
+
...
+
sin
(
n
−
1
)
a
=
2
sin
(
2
a
)
cos
a
(
2
a
)
−
cos
(
n
−
2
1
)
a
b) A point on the circumference of a wheel, which, remaining in a vertical plane, rolls along a horizontal path, describes, at one revolution of the wheel, a curve having a length equal to four times the diameter of the wheel. Prove this by first considering tilting a regular
n
n
n
-gon.[hide=original wording for part b]Een punt van de omtrek van een wiel dat, in een verticaal vlak blijvend, rolt over een horizontaal gedachte weg, beschrijft bij één omwenteling van het wiel een kromme die een lengte heeft die gelijk is aan viermaal de middellijn van het wiel. Bewijs dit door eerst een rondkantelende regelmatige n-hoek te beschouwen.
2
1
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x^2 + y^2 = z^n diophantine
Prove that for all
n
∈
N
n \in N
n
∈
N
,
x
2
+
y
2
=
z
n
x^2 + y^2 = z^n
x
2
+
y
2
=
z
n
has solutions with
x
,
y
,
z
∈
N
x,y,z \in N
x
,
y
,
z
∈
N
.
1
1
Hide problems
min n, such that n = (a - 1) mod $a$ for all a \in {2,3,..., 10 }
Determine the smallest
n
n
n
such that
n
≡
(
a
−
1
)
n \equiv (a - 1)
n
≡
(
a
−
1
)
mod
a
a
a
for all
a
∈
{
2
,
3
,
.
.
.
,
10
}
a \in \{2,3,..., 10\}
a
∈
{
2
,
3
,
...
,
10
}
.