MathDB
Problems
Contests
National and Regional Contests
Netherlands Contests
Dutch Mathematical Olympiad
1988 Dutch Mathematical Olympiad
1988 Dutch Mathematical Olympiad
Part of
Dutch Mathematical Olympiad
Subcontests
(4)
4
1
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min max area of a square that has inscribed triangle of sides 2-3-3
Given is an isosceles triangle
A
B
C
ABC
A
BC
with
A
B
=
2
AB = 2
A
B
=
2
and
A
C
=
B
C
=
3
AC = BC = 3
A
C
=
BC
=
3
. We consider squares where
A
,
B
A, B
A
,
B
and
C
C
C
lie on the sides of the square (so not on the extension of such a side). Determine the maximum and minimum value of the area of such a square. Justify the answer.
3
1
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sum (1/a^n) = 1/(sum a^n) if sum 1/a=1/sum a
For certain
a
,
b
,
c
a,b,c
a
,
b
,
c
holds:
1
a
+
1
b
+
1
c
=
1
a
+
b
+
c
\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{1}{a+b+c}
a
1
+
b
1
+
c
1
=
a
+
b
+
c
1
Prove that for all odd
n
n
n
holds,
1
a
n
+
1
b
n
+
1
c
n
=
1
a
n
+
b
n
+
c
n
.
\frac{1}{a^n}+\frac{1}{b^n}+\frac{1}{c^n}=\frac{1}{a^n+b^n+c^n}.
a
n
1
+
b
n
1
+
c
n
1
=
a
n
+
b
n
+
c
n
1
.
1
1
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x_1^{-2}+x_2^{-2}+...+ x_n^{-2} =?
The real numbers
x
1
,
x
2
,
.
.
.
,
x
n
x_1,x_2,..., x_n
x
1
,
x
2
,
...
,
x
n
and
a
0
,
a
1
,
.
.
.
,
a
n
−
1
a_0,a_1,...,a_{n-1}
a
0
,
a
1
,
...
,
a
n
−
1
with
x
i
≠
0
x_i \ne 0
x
i
=
0
for
i
∈
{
1
,
2
,
.
.
,
n
}
i \in\{1,2,.., n\}
i
∈
{
1
,
2
,
..
,
n
}
are such that
(
x
−
x
1
)
(
x
−
x
2
)
.
.
.
(
x
−
x
n
)
=
x
n
+
a
n
−
1
x
n
−
1
+
.
.
.
+
a
1
x
+
a
0
(x-x_1)(x-x_2)...(x-x_n)=x^n+a_{n-1}x^{n-1}+...+a_1x+a_0
(
x
−
x
1
)
(
x
−
x
2
)
...
(
x
−
x
n
)
=
x
n
+
a
n
−
1
x
n
−
1
+
...
+
a
1
x
+
a
0
Express
x
1
−
2
+
x
2
−
2
+
.
.
.
+
x
n
−
2
x_1^{-2}+x_2^{-2}+...+ x_n^{-2}
x
1
−
2
+
x
2
−
2
+
...
+
x
n
−
2
in terms of
a
0
,
a
1
,
.
.
.
,
a
n
−
1
a_0,a_1,...,a_{n-1}
a
0
,
a
1
,
...
,
a
n
−
1
.
2
1
Hide problems
lim 2^{2n+1}(1-c_n) if C_{n+1}=\sqrt{(1+c_n)/2}
Given is a number
a
a
a
with 0
≤
α
≤
π
\le \alpha \le \pi
≤
α
≤
π
. A sequence
c
0
,
c
1
,
c
2
,
.
.
.
c_0,c_1, c_2,...
c
0
,
c
1
,
c
2
,
...
is defined as
c
0
=
cos
α
c_0=\cos \alpha
c
0
=
cos
α
C
n
+
1
=
1
+
c
n
2
f
o
r
n
=
0
,
1
,
2
,
.
.
.
C_{n+1}=\sqrt{\frac{1+c_n}{2}} \,\, for \,\,\, n=0,1,2,...
C
n
+
1
=
2
1
+
c
n
f
or
n
=
0
,
1
,
2
,
...
Calculate
lim
n
→
∞
2
2
n
+
1
(
1
−
c
n
)
\lim_{n\to \infty}2^{2n+1}(1-c_n)
lim
n
→
∞
2
2
n
+
1
(
1
−
c
n
)