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Problems
Contests
National and Regional Contests
Netherlands Contests
Dutch Mathematical Olympiad
1983 Dutch Mathematical Olympiad
1983 Dutch Mathematical Olympiad
Part of
Dutch Mathematical Olympiad
Subcontests
(4)
4
1
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equilateral triangle
Within an equilateral triangle of side
15
15
15
are
111
111
111
points. Prove that it is always possible to cover three of these points by a round coin of diameter
3
\sqrt{3}
3
, part of which may lie outside the triangle.
3
1
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real numbers
Suppose that
a
,
b
,
c
,
p
a,b,c,p
a
,
b
,
c
,
p
are real numbers with
a
,
b
,
c
a,b,c
a
,
b
,
c
not all equal, such that: a\plus{}\frac{1}{b}\equal{}b\plus{}\frac{1}{c}\equal{}c\plus{}\frac{1}{a}\equal{}p. Determine all possible values of
p
p
p
and prove that abc\plus{}p\equal{}0.
2
1
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last two digits
Prove that if
n
n
n
is an odd positive integer, then the last two digits of 2^{2n}(2^{2n\plus{}1}\minus{}1) in base
10
10
10
are
28
28
28
.
1
1
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divided triangle
A triangle
A
B
C
ABC
A
BC
can be divided into two isosceles triangles by a line through
A
A
A
. Given that one of the angles of the triangles is
3
0
∘
30^{\circ}
3
0
∘
, find all possible values of the other two angles.