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Problems
Contests
National and Regional Contests
Netherlands Contests
Dutch Mathematical Olympiad
1976 Dutch Mathematical Olympiad
1976 Dutch Mathematical Olympiad
Part of
Dutch Mathematical Olympiad
Subcontests
(5)
3
1
Hide problems
7 moves for king in the chessboard
In how many ways can the king in the chessboard reach the eighth rank in
7
7
7
moves from its original square on the first row?
5
1
Hide problems
f(k_0) < f(k), f(k) = k + [ n/k ] , k_0 = [ \sqrt{n} ] + 1
f
(
k
)
=
k
+
[
n
k
]
f(k) = k + \left[ \frac{n}{k}\right ]
f
(
k
)
=
k
+
[
k
n
]
,
k
∈
{
1
,
2
,
.
.
.
,
n
}
k \in \{1,2,..., n\}
k
∈
{
1
,
2
,
...
,
n
}
,
k
0
=
[
n
]
+
1
k_0 =\left[ \sqrt{n} \right] + 1
k
0
=
[
n
]
+
1
.Prove that
f
(
k
0
)
<
f
(
k
)
f(k_0) < f(k)
f
(
k
0
)
<
f
(
k
)
if
k
∈
{
1
,
2
,
.
.
.
,
n
}
k \in \{1,2,..., n\}
k
∈
{
1
,
2
,
...
,
n
}
1
1
Hide problems
8n + 7 is is never sum of three squares
Prove that there is no natural
n
n
n
such that
8
n
+
7
8n + 7
8
n
+
7
is the sum of three squares.
4
1
Hide problems
x^2 - (2a^2 + 4)x + a^2 + 2a + b = 0
For
a
,
b
,
x
∈
R
a,b, x \in R
a
,
b
,
x
∈
R
holds:
x
2
−
(
2
a
2
+
4
)
x
+
a
2
+
2
a
+
b
=
0
x^2 - (2a^2 + 4)x + a^2 + 2a + b = 0
x
2
−
(
2
a
2
+
4
)
x
+
a
2
+
2
a
+
b
=
0
. For which
b
b
b
does this equation have at least one root between
0
0
0
and
1
1
1
for all
a
a
a
?
2
1
Hide problems
AL, BM, CN concurrent and each bisect any other, 3 parallelograms
Given
△
A
B
C
\vartriangle ABC
△
A
BC
and a point
P
P
P
inside that triangle. The parallelograms
C
P
B
L
CPBL
CPB
L
,
A
P
C
M
APCM
A
PCM
and
B
P
A
N
BPAN
BP
A
N
are constructed. Prove that
A
L
AL
A
L
,
B
M
BM
BM
and
C
N
CN
CN
pass through one point
S
S
S
, and that
S
S
S
is the midpoint of
A
L
AL
A
L
,
B
M
BM
BM
and
C
N
CN
CN
.