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Problems
Contests
National and Regional Contests
Netherlands Contests
Dutch Mathematical Olympiad
2013 Dutch Mathematical Olympiad
2013 Dutch Mathematical Olympiad
Part of
Dutch Mathematical Olympiad
Subcontests
(5)
2
1
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3x3 system, x + y - z = -1 , x^2 - y^2 + z^2 = 1, - x^3 + y^3 + z^3 = -1
Find all triples
(
x
,
y
,
z
)
(x, y, z)
(
x
,
y
,
z
)
of real numbers satisfying:
x
+
y
−
z
=
−
1
x + y - z = -1
x
+
y
−
z
=
−
1
,
x
2
−
y
2
+
z
2
=
1
x^2 - y^2 + z^2 = 1
x
2
−
y
2
+
z
2
=
1
and
−
x
3
+
y
3
+
z
3
=
−
1
- x^3 + y^3 + z^3 = -1
−
x
3
+
y
3
+
z
3
=
−
1
1
1
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bw squares in a nxn table, largest possible value of n for a few conditions
In a table consisting of
n
n
n
by
n
n
n
small squares some squares are coloured black and the other squares are coloured white. For each pair of columns and each pair of rows the four squares on the intersections of these rows and columns must not all be of the same colour. What is the largest possible value of
n
n
n
?
4
1
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equations with product of positive divisors P(n) = 15n, P(n) = 15n^2
For a positive integer n the number
P
(
n
)
P(n)
P
(
n
)
is the product of the positive divisors of
n
n
n
. For example,
P
(
20
)
=
8000
P(20) = 8000
P
(
20
)
=
8000
, as the positive divisors of
20
20
20
are
1
,
2
,
4
,
5
,
10
1, 2, 4, 5, 10
1
,
2
,
4
,
5
,
10
and
20
20
20
, whose product is
1
⋅
2
⋅
4
⋅
5
⋅
10
⋅
20
=
8000
1 \cdot 2 \cdot 4 \cdot 5 \cdot 10 \cdot 20 = 8000
1
⋅
2
⋅
4
⋅
5
⋅
10
⋅
20
=
8000
. (a) Find all positive integers
n
n
n
satisfying
P
(
n
)
=
15
n
P(n) = 15n
P
(
n
)
=
15
n
. (b) Show that there exists no positive integer
n
n
n
such that
P
(
n
)
=
15
n
2
P(n) = 15n^2
P
(
n
)
=
15
n
2
.
5
1
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digit ''5'' in the result of sum 1 + 10 + 19 + 28 + 37 +...+ 10^{2013}
The number
S
S
S
is the result of the following sum:
1
+
10
+
19
+
28
+
37
+
.
.
.
+
1
0
2013
1 + 10 + 19 + 28 + 37 +...+ 10^{2013}
1
+
10
+
19
+
28
+
37
+
...
+
1
0
2013
If one writes down the number
S
S
S
, how often does the digit `
5
5
5
' occur in the result?
3
1
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angle chasing in a quadrilateral with 2 parallel sides, few equal segments
The sides
B
C
BC
BC
and
A
D
AD
A
D
of a quadrilateral
A
B
C
D
ABCD
A
BC
D
are parallel and the diagonals intersect in
O
O
O
. For this quadrilateral
∣
C
D
∣
=
∣
A
O
∣
|CD| =|AO|
∣
C
D
∣
=
∣
A
O
∣
and
∣
B
C
∣
=
∣
O
D
∣
|BC| = |OD|
∣
BC
∣
=
∣
O
D
∣
hold. Furthermore
C
A
CA
C
A
is the angular bisector of angle
B
C
D
BCD
BC
D
. Determine the size of angle
A
B
C
ABC
A
BC
.[asy] unitsize(1 cm);pair A, B, C, D, O;D = (0,0); B = 3*dir(180 + 72); C = 3*dir(180 + 72 + 36); A = extension(D, D + (1,0), C, C + dir(180 - 36)); O = extension(A, C, B, D);draw(A--B--C--D--cycle); draw(B--D); draw(A--C);dot("
A
A
A
", A, N); dot("
B
B
B
", B, SW); dot("
C
C
C
", C, SE); dot("
D
D
D
", D, N); dot("
O
O
O
", O, E); [/asy]Attention: the figure is not drawn to scale.