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Problems
Contests
National and Regional Contests
Netherlands Contests
Dutch Mathematical Olympiad
1998 Dutch Mathematical Olympiad
1998 Dutch Mathematical Olympiad
Part of
Dutch Mathematical Olympiad
Subcontests
(5)
5
1
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Solve the equation
Find all real solutions of the following equation:
(
x
+
1995
)
(
x
+
1997
)
(
x
+
1999
)
(
x
+
2001
)
+
16
=
0.
(x + 1995)(x + 1997)(x + 1999)(x + 2001) + 16 = 0.
(
x
+
1995
)
(
x
+
1997
)
(
x
+
1999
)
(
x
+
2001
)
+
16
=
0.
4
1
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Perpendicular diagonals
Let
A
B
C
D
ABCD
A
BC
D
be a convex quadrilateral such that
A
C
⊥
B
D
AC \perp BD
A
C
⊥
B
D
. (a) Prove that
A
B
2
+
C
D
2
=
B
C
2
+
D
A
2
AB^2 + CD^2 = BC^2 + DA^2
A
B
2
+
C
D
2
=
B
C
2
+
D
A
2
. (b) Let
P
Q
R
S
PQRS
PQRS
be a convex quadrilateral such that
P
Q
=
A
B
PQ = AB
PQ
=
A
B
,
Q
R
=
B
C
QR = BC
QR
=
BC
,
R
S
=
C
D
RS = CD
RS
=
C
D
and
S
P
=
D
A
SP = DA
SP
=
D
A
. Prove that
P
R
⊥
Q
S
PR \perp QS
PR
⊥
QS
.
3
1
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Least common multiple
Let
m
m
m
and
n
n
n
be positive integers such that
m
−
n
=
189
m - n = 189
m
−
n
=
189
and such that the least common multiple of
m
m
m
and
n
n
n
is equal to
133866
133866
133866
. Find
m
m
m
and
n
n
n
.
2
1
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Volume of pyramid
Let
T
A
B
C
D
TABCD
T
A
BC
D
be a pyramid with top vertex
T
T
T
, such that its base
A
B
C
D
ABCD
A
BC
D
is a square of side length 4. It is given that, among the triangles
T
A
B
TAB
T
A
B
,
T
B
C
TBC
TBC
,
T
C
D
TCD
TC
D
and
T
D
A
TDA
T
D
A
, one can find an isosceles triangle and a right-angled triangle. Find all possible values for the volume of the pyramid.
1
1
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Permutation of 0, 1, 2, ..., 9
Consider any permutation
σ
\sigma
σ
of
{
0
,
1
,
2
,
…
,
9
}
\{0,1,2,\dots,9\}
{
0
,
1
,
2
,
…
,
9
}
and for each of the 8 triples of consecutive numbers in this permutation, consider the sum of these three numbers. Let
M
(
σ
)
M(\sigma)
M
(
σ
)
be the largest of these 8 sums. (For example, for the permutation
σ
=
(
4
,
6
,
2
,
9
,
0
,
1
,
8
,
5
,
7
,
3
)
\sigma = (4, 6, 2, 9, 0, 1, 8, 5, 7, 3)
σ
=
(
4
,
6
,
2
,
9
,
0
,
1
,
8
,
5
,
7
,
3
)
we get the 8 sums 12, 17, 11, 10, 9, 14, 20, 15, and
M
(
σ
)
=
20
M(\sigma) = 20
M
(
σ
)
=
20
.) (a) Find a permutation
σ
1
\sigma_1
σ
1
such that
M
(
σ
1
)
=
13
M(\sigma_1) = 13
M
(
σ
1
)
=
13
. (b) Does there exist a permutation
σ
2
\sigma_2
σ
2
such that
M
(
σ
2
)
=
12
M(\sigma_2) = 12
M
(
σ
2
)
=
12
?