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Problems
Contests
National and Regional Contests
Netherlands Contests
Dutch Mathematical Olympiad
1967 Dutch Mathematical Olympiad
1967 Dutch Mathematical Olympiad
Part of
Dutch Mathematical Olympiad
Subcontests
(5)
1
1
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PAB and PCD have equal areas in convex ABCD
In this exercise we only consider convex quadrilaterals.(a) For such a quadrilateral
A
B
C
D
ABCD
A
BC
D
, determine the set of points
P
P
P
contained within that quadrilateral for which
P
A
PA
P
A
and
P
C
PC
PC
divide the quadrilateral into two pieces of equal areas.(b) Prove that there is a point
P
P
P
inside such a quadrilateral, such that the triangles
P
A
B
PAB
P
A
B
and
P
C
D
PCD
PC
D
have equal areas, as well as the triangles
P
B
C
PBC
PBC
and
P
A
D
PAD
P
A
D
.(c) Find out which quadrilaterals
A
B
C
D
ABCD
A
BC
D
contains a point
P
P
P
, so that the triangles
P
A
B
PAB
P
A
B
,
P
B
C
PBC
PBC
,
P
C
D
PCD
PC
D
and
P
D
A
PDA
P
D
A
have equal areas.
3
1
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EA // diagonal if AB,BC,CD, DE // one diagonal, in convex ABCDE
The convex pentagon
A
B
C
D
E
ABC DE
A
BC
D
E
is given, such that
A
B
,
B
C
,
C
D
AB,BC,CD
A
B
,
BC
,
C
D
and
D
E
DE
D
E
are parallel to one of the diagonals. Prove that this also applies to
E
A
EA
E
A
.
5
1
Hide problems
sequence ([n x])
Consider rows of the form:
[
x
]
,
[
2
x
]
,
[
3
x
]
,
.
.
.
[x], [2x], [3x], ...
[
x
]
,
[
2
x
]
,
[
3
x
]
,
...
Proof that, if
N
∈
N
N \in N
N
∈
N
does not occur in the sequence
(
[
n
x
]
)
([n x])
([
n
x
])
, then there is an
n
∈
N
n \in N
n
∈
N
with
n
−
1
<
N
x
<
n
−
1
x
n - 1 < \frac{N}{x}< n -\frac{1}{x}
n
−
1
<
x
N
<
n
−
x
1
Prove that, for
x
,
y
∉
Q
x, y \notin Q
x
,
y
∈
/
Q
:
1
x
+
1
y
=
1
\frac{1}{x}+\frac{1}{y} = 1
x
1
+
y
1
=
1
, then each
N
∈
N
N \in N
N
∈
N
term is either of
(
[
n
x
]
)
([nx])
([
n
x
])
or of
(
[
n
y
]
)
([ny])
([
n
y
])
.
2
1
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arithmetic sequence with infinite terms perfect cubes
Consider arithmetic sequences where all terms are natural numbers. If the first term of such a sequence is
1
1
1
, prove that that sequence contains infinitely many terms that are the cube of a natural number. Give an example of such a sequence in which no term is the cube of a natural number and show the correctness of this example.
4
1
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ac + bd > ab if a, b, c, d > 0 with a + b < c + d
The following applies:
a
,
b
,
c
,
d
>
0
,
a
+
b
<
c
+
d
a, b, c, d > 0 , a + b < c + d
a
,
b
,
c
,
d
>
0
,
a
+
b
<
c
+
d
Prove that
a
c
+
b
d
>
a
b
.
ac + bd > ab.
a
c
+
b
d
>
ab
.