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Problems
Contests
National and Regional Contests
Netherlands Contests
Dutch Mathematical Olympiad
1985 Dutch Mathematical Olympiad
1985 Dutch Mathematical Olympiad
Part of
Dutch Mathematical Olympiad
Subcontests
(4)
4
1
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concurrent diagonals
A convex hexagon
A
B
C
D
E
F
ABCDEF
A
BC
D
EF
is such that each of the diagonals
A
D
,
B
E
,
C
F
AD,BE,CF
A
D
,
BE
,
CF
divides the hexagon into two parts of equal area. Prove that these three diagonals are concurrent.
3
1
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factory
In a factory, square tables of
40
×
40
40 \times 40
40
×
40
are tiled with four tiles of size
20
×
20
20 \times 20
20
×
20
. All tiles are the same and decorated in the same way with an asymmetric pattern such as the letter
J
J
J
. How many different types of tables can be produced in this way?
2
1
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squares
Among the numbers 11n \plus{} 10^{10}, where
1
≤
n
≤
1
0
10
1 \le n \le 10^{10}
1
≤
n
≤
1
0
10
is an integer, how many are squares?
1
1
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determine four numbers
For some
p
p
p
, the equation x^3 \plus{} px^2 \plus{} 3x \minus{} 10 \equal{} 0 has three real solutions
a
,
b
,
c
a,b,c
a
,
b
,
c
such that c \minus{} b \equal{} b \minus{} a > 0. Determine
a
,
b
,
c
,
a,b,c,
a
,
b
,
c
,
and
p
p
p
.