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Contests
National and Regional Contests
Belgium Contests
Flanders Math Olympiad
2010 Flanders Math Olympiad
2010 Flanders Math Olympiad
Part of
Flanders Math Olympiad
Subcontests
(4)
4
1
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in snack bar Pita Goras, the owner checks his accounts
In snack bar Pita Goras, the owner checks his accounts. He writes on every line either a positive amount in case of an income or a negative amount in case of an expense. He says to his accountant, “If I change the amounts of random
5
5
5
adding consecutive lines, I always get a strictly positive result.” "Indeed," the accountant answers him, “but if you put the sums of
7
7
7
consecutive lines add up, you always get a strictly negative result.” How many lines are there maximum on his sheet?
1
1
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no of zeros 101^{100} - 1 ends with
How many zeros does
10
1
100
−
1
101^{100} - 1
10
1
100
−
1
end with?
3
1
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AB//CD when <B= 2<A , angle bisector of B intersects perp.bisector of AC at D
In a triangle
A
B
C
ABC
A
BC
,
∠
B
=
2
∠
A
≠
9
0
o
\angle B= 2\angle A \ne 90^o
∠
B
=
2∠
A
=
9
0
o
. The inner bisector of
B
B
B
intersects the perpendicular bisector of
[
A
C
]
[AC]
[
A
C
]
at a point
D
D
D
. Prove that
A
B
∥
C
D
AB \parallel CD
A
B
∥
C
D
.
2
1
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trisection of diagonal by perp. in # with 60^o (2010 VWO Flanders MO p3)
A parallelogram with an angle of
6
0
o
60^o
6
0
o
has
a
a
a
as the longest side and a shortest side
b
b
b
. Let's take the perpendiculars down from the vertices of the obtuse angles to the longest diagonal, then it is divided into three equal parts. Determine the ratio
a
b
\frac{a}{b}
b
a
.