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Problems
Contests
National and Regional Contests
Netherlands Contests
Dutch Mathematical Olympiad
1980 Dutch Mathematical Olympiad
1980 Dutch Mathematical Olympiad
Part of
Dutch Mathematical Olympiad
Subcontests
(4)
4
1
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six kinds of banknotes wanted
In Venetiania, the smallest currency is the ducat. The finance minister instructs his officials as follows: "I wish six kinds of banknotes, each worth a whole number of ducats. Those six values must be such that there exists a number N with the following property: Any amount of money of
n
n
n
ducats (
n
n
n
positive and integer) where
n
≤
N
n \le N
n
≤
N
may be paid in such a way that no more than two copies of each kind are required either to pay or to return. I also wish those six values to be as large as possible for
N
N
N
. Determine those six values and provide proof that all conditions have been met." Solve the problem of those officials
2
1
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product of all divisors of 1980^n
Find the product of all divisors of
198
0
n
1980^n
198
0
n
,
n
≥
1
n \ge 1
n
≥
1
.
1
1
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1/a < x_0<2/a , x_0 is zero with smallest abs. value f(x) = x^3-ax+1
f
(
x
)
=
x
3
−
a
x
+
1
f(x) = x^3-ax+1
f
(
x
)
=
x
3
−
a
x
+
1
,
a
∈
R
a \in R
a
∈
R
has three different zeros in
R
R
R
. Prove that for the zero
x
o
x_o
x
o
with the smallest absolute value holds:
1
a
<
x
0
<
2
a
\frac{1}{a}< x_0 < \frac{2}{a}
a
1
<
x
0
<
a
2
3
1
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concurrency of 3 perpendiculars passing through midpoints
Given is the non-right triangle
A
B
C
ABC
A
BC
.
D
,
E
D,E
D
,
E
and
F
F
F
are the feet of the respective altitudes from
A
,
B
A,B
A
,
B
and
C
C
C
.
P
,
Q
P,Q
P
,
Q
and
R
R
R
are the respective midpoints of the line segments
E
F
EF
EF
,
F
D
FD
F
D
and
D
E
DE
D
E
.
p
⊥
B
C
p \perp BC
p
⊥
BC
passes through
P
P
P
,
q
⊥
C
A
q \perp CA
q
⊥
C
A
passes through
Q
Q
Q
and
r
⊥
A
B
r \perp AB
r
⊥
A
B
passes through
R
R
R
. Prove that the lines
p
,
q
p, q
p
,
q
and
r
r
r
pass through one point.