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Problems
Contests
National and Regional Contests
Netherlands Contests
Dutch Mathematical Olympiad
2002 Dutch Mathematical Olympiad
2002 Dutch Mathematical Olympiad
Part of
Dutch Mathematical Olympiad
Subcontests
(5)
5
1
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<A=2<B, AB=3, AC=2, BC=?
In triangle
A
B
C
ABC
A
BC
, angle
A
A
A
is twice as large as angle
B
B
B
.
A
B
=
3
AB = 3
A
B
=
3
and
A
C
=
2
AC = 2
A
C
=
2
. Calculate
B
C
BC
BC
.
4
1
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5 pairs of cartoon characters around a round table of 10 chairs
Five pairs of cartoon characters, Donald and Katrien Duck, Asterix and Obelix, Suske and Wiske, Tom and Jerry, Heer Bommel and Tom Poes, sit around a round table with
10
10
10
chairs. The two members of each pair ensure that they sit next to each other. In how many different ways can the ten seats be occupied? Two ways are different if they cannot be transferred to each other by a rotation.
3
1
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triangle with lattice vertices and integer sidelenghts has even perimeter
A
,
B
A, B
A
,
B
and
C
C
C
are points in the plane with integer coordinates. The lengths of the sides of triangle
A
B
C
ABC
A
BC
are integer numbers. Prove that the perimeter of the triangle is an even number.
2
1
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diophantine (1+1/x)(1+1/y)(1+1/z) = 3
Determine all triplets
(
x
,
y
,
z
)
(x, y, z)
(
x
,
y
,
z
)
of positive integers with
x
≤
y
≤
z
x \le y \le z
x
≤
y
≤
z
that satisfy
(
1
+
1
x
)
(
1
+
1
y
)
(
1
+
1
z
)
=
3
\left(1+\frac1x \right)\left(1+\frac1y \right)\left(1+\frac1z \right) = 3
(
1
+
x
1
)
(
1
+
y
1
)
(
1
+
z
1
)
=
3
1
1
Hide problems
reflections inside a square, passing through vertices light beam escapes
The sides of a
10
10
10
by
10
10
10
square
A
B
C
D
ABCD
A
BC
D
are reflective on the inside. A beam of light enters the square via the vertex
A
A
A
and heads to the point
P
P
P
on
C
D
CD
C
D
with
C
P
=
3
CP = 3
CP
=
3
and
P
D
=
7
PD = 7
P
D
=
7
. In
P
P
P
it naturally reflects on the
C
D
CD
C
D
side. The light beam can only leave the square via one of the angular points
A
,
B
,
C
A, B, C
A
,
B
,
C
or
D
D
D
. What is the distance that the light beam travels within the square before it leaves the square again? By which vertex does that happen?