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Problems
Contests
National and Regional Contests
Netherlands Contests
Dutch Mathematical Olympiad
1968 Dutch Mathematical Olympiad
1968 Dutch Mathematical Olympiad
Part of
Dutch Mathematical Olympiad
Subcontests
(5)
1
1
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min <ACP when P lies on base AB of isosceles ABC with AP/PB=1/2
On the base
A
B
AB
A
B
of the isosceles triangle
A
B
C
ABC
A
BC
, lies the point
P
P
P
such that
A
P
:
P
B
=
1
:
2
AP : PB = 1 : 2
A
P
:
PB
=
1
:
2
. Determine the minimum of
∠
A
C
P
\angle ACP
∠
A
CP
.
3
1
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similar triangles with integer sides, and 2 equal sides
△
A
B
C
∼
△
A
′
B
′
C
′
\vartriangle ABC \sim \vartriangle A'B'C'
△
A
BC
∼
△
A
′
B
′
C
′
.
△
A
B
C
\vartriangle ABC
△
A
BC
has sides
a
,
b
,
c
a,b,c
a
,
b
,
c
and
△
A
′
B
′
C
′
\vartriangle A'B'C'
△
A
′
B
′
C
′
has sides
a
′
,
b
′
,
c
′
a',b',c'
a
′
,
b
′
,
c
′
. Two sides of
△
A
B
C
\vartriangle ABC
△
A
BC
are equal to sides of
△
A
′
B
′
C
′
\vartriangle A'B'C'
△
A
′
B
′
C
′
. Furthermore,
a
<
a
′
a < a'
a
<
a
′
,
a
<
b
<
c
a < b < c
a
<
b
<
c
,
a
=
8
a = 8
a
=
8
. Prove that there is exactly one pair of such triangles with all sides integers.
4
1
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line construction, which coincides with itself after reflections wrt 3 lines
Given is a triangle
A
B
C
ABC
A
BC
. A line
ℓ
\ell
ℓ
passes through reflection wrt
B
C
BC
BC
changes into the line
ℓ
′
\ell'
ℓ
′
,
ℓ
′
\ell'
ℓ
′
changes into
ℓ
′
′
\ell''
ℓ
′′
through reflection wrt
A
C
AC
A
C
and
ℓ
′
′
\ell''
ℓ
′′
through reflection wrt
A
B
AB
A
B
changes into
ℓ
′
′
′
\ell'''
ℓ
′′′
. Construct the line
ℓ
\ell
ℓ
given that
ℓ
′
′
′
\ell'''
ℓ
′′′
coincides with
ℓ
\ell
ℓ
.
5
1
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no of rectangles in n x n square
A square of side
n
n
n
(
n
n
n
natural) is divided into
n
2
n^2
n
2
squares of side
1
1
1
. Each pair of "horizontal" boundary lines and each pair of "vertical" boundary lines enclose a rectangle (a square is also considered a rectangle). A rectangle has a length and a width; the width is less than or equal to the length. (a) Prove that there are
8
8
8
rectangles of width
n
−
1
n - 1
n
−
1
. (b) Determine the number of rectangles with width
n
−
k
n -k
n
−
k
(
0
≤
k
≤
n
−
1
,
k
0\le k \le n -1,k
0
≤
k
≤
n
−
1
,
k
integer). (c) Determine a formula for
1
3
+
2
3
+
.
.
.
+
n
3
1^3 + 2^3 +...+ n^3
1
3
+
2
3
+
...
+
n
3
.
2
1
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1/2 ( N / a +a ) >= \sqrt{N}
It holds:
N
,
a
>
0
N,a > 0
N
,
a
>
0
. Prove that
1
2
(
N
a
+
a
)
≥
N
\frac12 \left(\frac{N}{a}+a \right) \ge \sqrt{N}
2
1
(
a
N
+
a
)
≥
N
, and if
N
≥
1
N \ge 1
N
≥
1
and
a
=
[
N
]
a = [\sqrt{N}]
a
=
[
N
]
. Prove that if
a
≠
N
:
1
2
(
N
a
+
a
)
a \ne \sqrt{N}: \frac12 \left(\frac{N}{a}+a \right)
a
=
N
:
2
1
(
a
N
+
a
)
is a better approximation for
N
\sqrt{N}
N
than
a
a
a
.