MathDB
Problems
Contests
National and Regional Contests
Netherlands Contests
Dutch Mathematical Olympiad
1965 Dutch Mathematical Olympiad
1965 Dutch Mathematical Olympiad
Part of
Dutch Mathematical Olympiad
Subcontests
(5)
2
1
Hide problems
5|(n + 6)^2+(n + 7)^2+ ... +(n + 10)^2 - [(n + 1)^2 + (n + 2)^2 +...+ (n + 5)]
Prove that
S
1
=
(
n
+
1
)
2
+
(
n
+
2
)
2
+
.
.
.
+
(
n
+
5
)
2
S_1 = (n + 1)^2 + (n + 2)^2 +...+ (n + 5)^2
S
1
=
(
n
+
1
)
2
+
(
n
+
2
)
2
+
...
+
(
n
+
5
)
2
is divisible by
5
5
5
for every
n
n
n
. Prove that for no
n
n
n
:
∑
ℓ
=
1
5
(
n
+
ℓ
)
2
\sum_{\ell=1}^5 (n+\ell)^2
∑
ℓ
=
1
5
(
n
+
ℓ
)
2
is a perfect square. Let
S
2
=
(
n
+
6
)
2
+
(
n
+
7
)
2
+
.
.
.
+
(
n
+
10
)
2
S_2=(n + 6)^2 + (n + 7)^2 + ... + (n + 10)^2
S
2
=
(
n
+
6
)
2
+
(
n
+
7
)
2
+
...
+
(
n
+
10
)
2
. Prove that
S
1
⋅
S
2
S_1 \cdot S_2
S
1
⋅
S
2
is divisible by
150
150
150
.
4
1
Hide problems
points on plane
We consider a number of points in a plane. Each of these points is connected to at least one of the other points by a line segment, in such a way that a figure arises that does not break up into different parts (that is, from any point along drawn line segments we can reach any other point).. We assign a point the ”order”
n
n
n
, when in this point
n
n
n
line segments meet. We characterize the obtained figure by writing down the order of each of its points one after the other. For example, a hexagon is characterized by the combination
{
2
,
2
,
2
,
2
,
2
,
2
}
\{2,2,2,2,2,2\}
{
2
,
2
,
2
,
2
,
2
,
2
}
and a star with six rays by
{
6
,
1
,
1
,
1
,
1
,
1
,
1
}
\{6,1,1,1,1,1,1\}
{
6
,
1
,
1
,
1
,
1
,
1
,
1
}
. (a) Sketch a figure' belonging to the combination
{
4
,
3
,
3
,
3
,
3
}
\{4,3,3,3,3\}
{
4
,
3
,
3
,
3
,
3
}
. (b) Give the combinations of all possible figures, of which the sum of the order numbers is equal to
6
6
6
. (c) Prove that every such combination contains an even number of odd numbers.
1
1
Hide problems
t_{n+1} = t_n x t_{n+2} , P_n = product of the first n terms
We consider the sequence
t
1
,
t
2
,
t
3
,
.
.
.
t_1,t_2,t_3,...
t
1
,
t
2
,
t
3
,
...
By
P
n
P_n
P
n
we mean the product of the first
n
n
n
terms of the sequence. Given that
t
n
+
1
=
t
n
⋅
t
n
+
2
t_{n+1} = t_n \cdot t_{n+2}
t
n
+
1
=
t
n
⋅
t
n
+
2
for each
n
n
n
, and that
P
40
=
P
80
=
8
P_{40} = P_{80} = 8
P
40
=
P
80
=
8
. Calculate
t
1
t_1
t
1
and
t
2
t_2
t
2
.
5
1
Hide problems
f(x+y)+f(x-y)=2f(x)+2f(y)
The function ƒ. which is defined for all real numbers satisfies:
f
(
x
+
y
)
+
f
(
x
−
y
)
=
2
f
(
x
)
+
2
f
(
y
)
f(x+y)+f(x-y)=2f(x)+2f(y)
f
(
x
+
y
)
+
f
(
x
−
y
)
=
2
f
(
x
)
+
2
f
(
y
)
Prove that
f
(
0
)
=
0
f(0) = 0
f
(
0
)
=
0
,
f
(
−
x
)
=
f
(
x
)
f(-x) = f(x)
f
(
−
x
)
=
f
(
x
)
,
f
(
2
x
)
=
4
f
(
x
)
f(2x) = 4 f (x)
f
(
2
x
)
=
4
f
(
x
)
,
f
(
x
+
y
+
z
)
=
f
(
x
+
y
)
+
f
(
y
+
z
)
+
f
(
z
+
x
)
−
f
(
x
)
−
f
(
y
)
−
f
(
z
)
.
f(x + y + z) = f(x + y) + f(y + z) + f(z + x) -f(x) - f(y) -f(z).
f
(
x
+
y
+
z
)
=
f
(
x
+
y
)
+
f
(
y
+
z
)
+
f
(
z
+
x
)
−
f
(
x
)
−
f
(
y
)
−
f
(
z
)
.
3
1
Hide problems
number of points C on a line such that ABC is isosceles for fixed A,B
Given are the points
A
A
A
and
B
B
B
in the plane. If
x
x
x
is a straight line is in that plane, and
x
x
x
does not coincide with the perpendicular bisectror of
A
B
AB
A
B
, then denote the number of points
C
C
C
located at
x
x
x
such that
△
A
B
C
\vartriangle ABC
△
A
BC
is isosceles, as the "weight of the line
x
x
x
”. Prove that the weight of any line
x
x
x
is at most
5
5
5
and determine the set of points
P
P
P
which has a line with weight
1
1
1
, but none with weight
0
0
0
.