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Problems
Contests
National and Regional Contests
Netherlands Contests
Dutch Mathematical Olympiad
1973 Dutch Mathematical Olympiad
1973 Dutch Mathematical Olympiad
Part of
Dutch Mathematical Olympiad
Subcontests
(5)
1
1
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inscribed trriangle with min perimeter in a triangle with angle 60^o
Given is a triangle
A
B
C
ABC
A
BC
,
∠
C
=
6
0
o
\angle C = 60^o
∠
C
=
6
0
o
,
R
R
R
the midpoint of side
A
B
AB
A
B
. There exist a point
P
P
P
on the line
B
C
BC
BC
and a point
Q
Q
Q
on the line
A
C
AC
A
C
such that the perimeter of the triangle
P
Q
R
PQR
PQR
is minimal. a) Prove that and also indicate how the points
P
P
P
and
Q
Q
Q
can be constructed. b) If
A
B
=
c
AB = c
A
B
=
c
,
A
C
=
b
AC = b
A
C
=
b
,
B
C
=
a
BC = a
BC
=
a
, then prove that the perimeter of the triangle
P
Q
R
PQR
PQR
equals
1
2
3
c
2
+
6
a
b
\frac12\sqrt{3c^2+6ab}
2
1
3
c
2
+
6
ab
.
5
1
Hide problems
a_{n+1} = a_n - n if a_n >= n , or a_{n+1} = a_n + n if a_n < n
An infinite sequence of integers
a
1
,
a
2
,
a
3
,
.
.
.
a_1,a_2,a_3, ...
a
1
,
a
2
,
a
3
,
...
is given with
a
1
=
0
a_1 = 0
a
1
=
0
and further holds for every natural number
n
n
n
that
a
n
+
1
=
a
n
−
n
a_{n+1} = a_n - n
a
n
+
1
=
a
n
−
n
if
a
n
≥
n
a_n \ge n
a
n
≥
n
and
a
n
+
1
=
a
n
+
n
a_{n+1} = a_n + n
a
n
+
1
=
a
n
+
n
if
a
n
<
n
a_n < n
a
n
<
n
. (a) Prove that there are infinitely many numbers in the sequence equal to
0
0
0
. (b) Express in terms of
k
k
k
the ordinal number of the
k
e
k^e
k
e
number from the sequence, which is equal to
0
0
0
.
4
1
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x_{n+1} = \sqrt{x_n -\frac14}
We have an infinite sequence of real numbers
x
0
,
x
1
,
x
2
,
.
.
.
x_0,x_1, x_2, ...
x
0
,
x
1
,
x
2
,
...
such that
x
n
+
1
=
x
n
−
1
4
x_{n+1} = \sqrt{x_n -\frac14}
x
n
+
1
=
x
n
−
4
1
holds for all natural
n
n
n
and moreover
x
0
∈
1
2
x_0 \in \frac12
x
0
∈
2
1
. (a) Prove that for every natural
n
n
n
holds:
x
n
>
1
2
x_n > \frac12
x
n
>
2
1
(b) Prove that
lim
n
→
∞
x
n
\lim_{n \to \infty} x_n
lim
n
→
∞
x
n
exists. Calculate this limit.
3
1
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<PCA=? <PAB = 30^o, <PBA = 20^o in 100-40-40 triangle
The angles
A
A
A
and
B
B
B
of base of the isosceles triangle
A
B
C
ABC
A
BC
are equal to
4
0
o
40^o
4
0
o
. Inside
△
A
B
C
\vartriangle ABC
△
A
BC
,
P
P
P
is such that
∠
P
A
B
=
3
0
o
\angle PAB = 30^o
∠
P
A
B
=
3
0
o
and
∠
P
B
A
=
2
0
o
\angle PBA = 20^o
∠
PB
A
=
2
0
o
. Calculate, without table,
∠
P
C
A
\angle PCA
∠
PC
A
.
2
1
Hide problems
exactly one sequence of 2n + 1 consecutive numbers
Prove that for every
n
∈
N
n \in N
n
∈
N
there exists exactly one sequence of
2
n
+
1
2n + 1
2
n
+
1
consecutive numbers, such that the sum of the squares of the first
n
+
1
n+1
n
+
1
numbers is equal to the sum of the squares of the last
n
n
n
numbers. Also express the smallest number of that sequence in terms of
n
n
n
.