MathDB
Problems
Contests
National and Regional Contests
Netherlands Contests
Dutch Mathematical Olympiad
1999 Dutch Mathematical Olympiad
1999 Dutch Mathematical Olympiad
Part of
Dutch Mathematical Olympiad
Subcontests
(5)
5
1
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a[n] = n^2 + c
Let
c
c
c
be a nonnegative integer, and define
a
n
=
n
2
+
c
a_n = n^2 + c
a
n
=
n
2
+
c
(for
n
≥
1
)
n \geq 1)
n
≥
1
)
. Define
d
n
d_n
d
n
as the greatest common divisor of
a
n
a_n
a
n
and
a
n
+
1
a_{n + 1}
a
n
+
1
. (a) Suppose that
c
=
0
c = 0
c
=
0
. Show that
d
n
=
1
,
∀
n
≥
1
d_n = 1,\ \forall n \geq 1
d
n
=
1
,
∀
n
≥
1
. (b) Suppose that
c
=
1
c = 1
c
=
1
. Show that
d
n
∈
{
1
,
5
}
,
∀
n
≥
1
d_n \in \{1,5\},\ \forall n \geq 1
d
n
∈
{
1
,
5
}
,
∀
n
≥
1
. (c) Show that
d
n
≤
4
c
+
1
,
∀
n
≥
1
d_n \leq 4c + 1,\ \forall n \geq 1
d
n
≤
4
c
+
1
,
∀
n
≥
1
.
4
1
Hide problems
8 x 8 matrix containing positive integers
Consider a matrix of size
8
×
8
8 \times 8
8
×
8
, containing positive integers only. One may repeatedly transform the entries of the matrix according to the following rules: -Multiply all entries in some row by 2. -Subtract 1 from all entries in some column. Prove that one can transform the given matrix into the zero matrix.
3
1
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Rotating a square + Projection on a line
Let
A
B
C
D
ABCD
A
BC
D
be a square and let
ℓ
\ell
ℓ
be a line. Let
M
M
M
be the centre of the square. The diagonals of the square have length 2 and the distance from
M
M
M
to
ℓ
\ell
ℓ
exceeds 1. Let
A
′
,
B
′
,
C
′
,
D
′
A',B',C',D'
A
′
,
B
′
,
C
′
,
D
′
be the orthogonal projections of
A
,
B
,
C
,
D
A,B,C,D
A
,
B
,
C
,
D
onto
ℓ
\ell
ℓ
. Suppose that one rotates the square, such that
M
M
M
is invariant. The positions of
A
,
B
,
C
,
D
,
A
′
,
B
′
,
C
′
,
D
′
A,B,C,D,A',B',C',D'
A
,
B
,
C
,
D
,
A
′
,
B
′
,
C
′
,
D
′
change. Prove that the value of
A
A
′
2
+
B
B
′
2
+
C
C
′
2
+
D
D
′
2
AA'^2 + BB'^2 + CC'^2 + DD'^2
A
A
′2
+
B
B
′2
+
C
C
′2
+
D
D
′2
does not change.
2
1
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Colouring of a 9 x 9 square
A
9
×
9
9 \times 9
9
×
9
square consists of
81
81
81
unit squares. Some of these unit squares are painted black, and the others are painted white, such that each
2
×
3
2 \times 3
2
×
3
rectangle and each
3
×
2
3 \times 2
3
×
2
rectangle contain exactly 2 black unit squares and 4 white unit squares. Determine the number of black unit squares.
1
1
Hide problems
Function with f(mn) = f(m)f(n)
Let
f
:
Z
→
{
−
1
,
1
}
f: \mathbb{Z} \rightarrow \{-1,1\}
f
:
Z
→
{
−
1
,
1
}
be a function such that
f
(
m
n
)
=
f
(
m
)
f
(
n
)
,
∀
m
,
n
∈
Z
.
f(mn) =f(m)f(n),\ \forall m,n \in \mathbb{Z}.
f
(
mn
)
=
f
(
m
)
f
(
n
)
,
∀
m
,
n
∈
Z
.
Show that there exists a positive integer
a
a
a
such that
1
≤
a
≤
12
1 \leq a \leq 12
1
≤
a
≤
12
and
f
(
a
)
=
f
(
a
+
1
)
=
1
f(a) = f(a + 1) = 1
f
(
a
)
=
f
(
a
+
1
)
=
1
.