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Problems
Contests
National and Regional Contests
Netherlands Contests
Dutch Mathematical Olympiad
1992 Dutch Mathematical Olympiad
1992 Dutch Mathematical Olympiad
Part of
Dutch Mathematical Olympiad
Subcontests
(5)
5
1
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sequence
We consider regular
n
n
n
-gons with a fixed circumference
4
4
4
. Let
r
n
r_n
r
n
and
a
n
a_n
a
n
respectively be the distances from the center of such an
n
n
n
-gon to a vertex and to an edge.
(
a
)
(a)
(
a
)
Determine
a
4
,
r
4
,
a
8
,
r
8
a_4,r_4,a_8,r_8
a
4
,
r
4
,
a
8
,
r
8
.
(
b
)
(b)
(
b
)
Give an appropriate interpretation for
a
2
a_2
a
2
and
r
2
r_2
r
2
(
c
)
(c)
(
c
)
Prove that a_{2n}\equal{}\frac{1}{2} (a_n\plus{}r_n) and r_{2n}\equal{}\sqrt{a_2n r_n}.
(
d
)
(d)
(
d
)
Define u_0\equal{}0, u_1\equal{}1 and u_n\equal{}\frac{1}{2}(u_{n\minus{}2}\plus{}u_{n\minus{}1}) for
n
n
n
even or u_n\equal{}\sqrt{u_{n\minus{}2} u_{n\minus{}1}} for
n
n
n
odd. Determine
lim
n
→
∞
u
n
\displaystyle\lim_{n\to\infty}u_n
n
→
∞
lim
u
n
.
4
1
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weird inequality
For every positive integer
n
n
n
, we define
n
?
n?
n
?
as 1?\equal{}1 and n?\equal{}\frac{n}{(n\minus{}1)?} for
n
≥
2
n \ge 2
n
≥
2
. Prove that
1992
<
1992
?
<
4
3
1992
.
\sqrt{1992}<1992?<\frac{4}{3} \sqrt{1992}.
1992
<
1992
?
<
3
4
1992
.
3
1
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configuration
Consider the configuration of six squares as shown on the picture. Prove that the sum of the area of the three outer squares (
I
,
I
I
I,II
I
,
II
and
I
I
I
III
III
) equals three times the sum of the areas of the three inner squares (
I
V
,
V
IV,V
I
V
,
V
and
V
I
VI
V
I
).
2
1
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fraction
In the fraction below and its decimal notation (with period of length
4
4
4
) every letter represents a digit, and different letters denote different digits. The numerator and denominator are coprime. Determine the value of the fraction: \frac{ADA}{KOK}\equal{}0.SNELSNELSNELSNEL...
N
o
t
e
.
Note.
N
o
t
e
.
Ada Kok is a famous dutch swimmer, and "snel" is Dutch for "fast".
1
1
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probability
Four dice are thrown. What is the probability that the product of the number equals
36
?
36?
36
?