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Problems
Contests
National and Regional Contests
Netherlands Contests
Dutch Mathematical Olympiad
1993 Dutch Mathematical Olympiad
1993 Dutch Mathematical Olympiad
Part of
Dutch Mathematical Olympiad
Subcontests
(5)
5
1
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eleven points
Eleven distinct points
P
1
,
P
2
,
.
.
.
,
P
11
P_1,P_2,...,P_{11}
P
1
,
P
2
,
...
,
P
11
are given on a line so that
P
i
P
j
≤
1
P_i P_j \le 1
P
i
P
j
≤
1
for every
i
,
j
i,j
i
,
j
. Prove that the sum of all distances
P
i
P
j
,
1
≤
i
<
j
≤
11
P_i P_j, 1 \le i <j \le 11
P
i
P
j
,
1
≤
i
<
j
≤
11
, is smaller than
30
30
30
.
4
1
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circle
Let
C
C
C
be a circle with center
M
M
M
in a plane
V
V
V
, and
P
P
P
be a point not on the circle
C
C
C
.
(
a
)
(a)
(
a
)
If
P
P
P
is fixed, prove that AP^2\plus{}BP^2 is a constant for every diameter
A
B
AB
A
B
of the circle
C
C
C
.
(
b
)
(b)
(
b
)
Let
A
B
AB
A
B
be a fixed diameter of
C
C
C
and
P
P
P
a point on a fixed sphere
S
S
S
not intersecting
V
V
V
. Determine the points
P
P
P
on
S
S
S
that minimize AP^2\plus{}BP^2.
3
1
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sequence
A sequence of numbers is defined by u_1\equal{}a, u_2\equal{}b and u_{n\plus{}1}\equal{}\frac{u_n\plus{}u_{n\minus{}1}}{2} for
n
≥
2
n \ge 2
n
≥
2
. Prove that
lim
n
→
∞
u
n
\displaystyle\lim_{n\to\infty}u_n
n
→
∞
lim
u
n
exists and express its value in terms of
a
a
a
and
b
b
b
.
2
1
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determine the ratio
In a triangle
A
B
C
ABC
A
BC
with \angle A\equal{}90^{\circ},
D
D
D
is the midpoint of
B
C
BC
BC
,
F
F
F
that of
A
B
AB
A
B
,
E
E
E
that of
A
F
AF
A
F
and
G
G
G
that of
F
B
FB
FB
. Segment
A
D
AD
A
D
intersects
C
E
,
C
F
CE,CF
CE
,
CF
and
C
G
CG
CG
in
P
,
Q
P,Q
P
,
Q
and
R
R
R
, respectively. Determine the ratio:
P
Q
Q
R
\frac{PQ}{QR}
QR
PQ
.
1
1
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set
Show that any subset of V\equal{} \{ 1,2,...,24,25 \} with
17
17
17
or more elements contains at least two distinct numbers the product of which is a perfect square.