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Problems
Contests
National and Regional Contests
Netherlands Contests
Dutch Mathematical Olympiad
1994 Dutch Mathematical Olympiad
1994 Dutch Mathematical Olympiad
Part of
Dutch Mathematical Olympiad
Subcontests
(5)
5
1
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easy inequality (old problem)
Three real numbers
a
,
b
,
c
a,b,c
a
,
b
,
c
satisfy the inequality |ax^2\plus{}bx\plus{}c| \le 1 for all x \in [\minus{}1,1]. Prove that |cx^2\plus{}bx\plus{}a| \le 2 for all x \in [\minus{}1,1].
4
1
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same areas
Let
P
P
P
be a point on the diagonal
B
D
BD
B
D
of a rectangle
A
B
C
D
ABCD
A
BC
D
,
F
F
F
be the projection of
P
P
P
on
B
C
BC
BC
, and H \not\equal{} B be the point on
B
C
BC
BC
such that BF\equal{}FH. If lines
P
C
PC
PC
and
A
H
AH
A
H
intersect at
Q
Q
Q
, prove that the areas of triangles
A
P
Q
APQ
A
PQ
and
C
H
Q
CHQ
C
H
Q
are equal.
3
1
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existence
(
a
)
(a)
(
a
)
Prove that every multiple of
6
6
6
can be written as a sum of four cubes.
(
b
)
(b)
(
b
)
Prove that every integer can be written as a sum of five cubes.
2
1
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sequence
A sequence of integers
a
1
,
a
2
,
a
3
,
.
.
.
a_1,a_2,a_3,...
a
1
,
a
2
,
a
3
,
...
is such that a_1\equal{}2, a_2\equal{}3, and a_{n\plus{}1}\equal{}2a_{n\minus{}1} or 3a_n\minus{}2a_{n\minus{}1} for all
n
≥
2
n \ge 2
n
≥
2
. Prove that no number between
1600
1600
1600
and
2000
2000
2000
can be an element of the sequence.
1
1
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rectangle
A unit square is divided into two rectangles in such a way that the smaller rectangle can be put on the greater rectangle with every vertex of the smaller on exactly one of the edges of the greater. Calculate the dimensions of the smaller rectangle.