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Problems
Contests
National and Regional Contests
Netherlands Contests
Dutch Mathematical Olympiad
1990 Dutch Mathematical Olympiad
1990 Dutch Mathematical Olympiad
Part of
Dutch Mathematical Olympiad
Subcontests
(4)
4
1
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regular 7-gon
If
A
B
C
D
E
F
G
ABCDEFG
A
BC
D
EFG
is a regular
7
7
7
-gon with side
1
1
1
, show that: \frac{1}{AC}\plus{}\frac{1}{AD}\equal{}1.
3
1
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polynomial
A polynomial f(x)\equal{}ax^4\plus{}bx^3\plus{}cx^2\plus{}dx with
a
,
b
,
c
,
d
>
0
a,b,c,d>0
a
,
b
,
c
,
d
>
0
is such that
f
(
x
)
f(x)
f
(
x
)
is an integer for x \in \{ \minus{}2,\minus{}1,0,1,2 \} and f(1)\equal{}1 and f(5)\equal{}70.
(
a
)
(a)
(
a
)
Show that a\equal{}\frac{1}{24}, b\equal{}\frac{1}{4},c\equal{}\frac{11}{24},d\equal{}\frac{1}{4}.
(
b
)
(b)
(
b
)
Prove that
f
(
x
)
f(x)
f
(
x
)
is an integer for all
x
∈
Z
x \in \mathbb{Z}
x
∈
Z
.
2
1
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sequence
Consider the sequence a_1\equal{}\frac{3}{2}, a_{n\plus{}1}\equal{}\frac{3a_n^2\plus{}4a_n\minus{}3}{4a_n^2}.
(
a
)
(a)
(
a
)
Prove that
1
<
a
n
1<a_n
1
<
a
n
and a_{n\plus{}1}
n
n
n
.
(
b
)
(b)
(
b
)
From
(
a
)
(a)
(
a
)
it follows that
lim
n
→
∞
a
n
\displaystyle\lim_{n\to\infty}a_n
n
→
∞
lim
a
n
exists. Find this limit.
(
c
)
(c)
(
c
)
Determine
lim
n
→
∞
a
1
a
2
a
3
.
.
.
a
n
\displaystyle\lim_{n\to\infty}a_1a_2a_3...a_n
n
→
∞
lim
a
1
a
2
a
3
...
a
n
.
1
1
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inequality in integers
Prove that for every integer n>1, 1 \cdot 3 \cdot 5 \cdot ... \cdot (2n\minus{}1)