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Contests
National and Regional Contests
Belgium Contests
Flanders Math Olympiad
2019 Flanders Math Olympiad
2019 Flanders Math Olympiad
Part of
Flanders Math Olympiad
Subcontests
(4)
4
1
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Knights of the Round Table , 34 chairs
The Knights of the Round Table are gathering. Around the table are
34
34
34
chairs, numbered from 1 to
34
34
34
. When everyone has sat down, it turns out that between every two knights there is a maximum of
r
r
r
places, which can be either empty or occupied by another knight. (a) For each
r
≤
15
r \le 15
r
≤
15
, determine the maximum number of knights present. (b) Determine for each
r
≤
15
r \le 15
r
≤
15
how many sets of occupied seats there are that match meet the given and where the maximum number of knights is present.
2
1
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sum of fractions whose numerator and denominator are positive divisors of 1000
Calculate the sum of all unsimplified fractions whose numerator and denominator are positive divisors of
1000
1000
1000
.
1
1
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\sqrt{d}=\sqrt{a}+\sqrt{b}, tangent balls in cylinder (2019 VWO Flanders MO p1)
Two touching balls with radii
a
a
a
and
b
b
b
are enclosed in a cylindrical tin of diameter
d
d
d
. Both balls hit the top surface and the shell of the cylinder. The largest ball also hits the bottom surface. Show that
d
=
a
+
b
\sqrt{d} =\sqrt{a} +\sqrt{b}
d
=
a
+
b
https://1.bp.blogspot.com/-O4B3P3bghFs/Xy1fDv9zGkI/AAAAAAAAMSQ/ePLVnsXsRi0mz3SWBpIzfGdsizWoLmGVACLcBGAsYHQ/s0/flanders%2B2019%2Bp1.png
3
1
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angle chasing starting with a 120-40-20 triangle ABC, | CP | = | AB | - | BC |
In triangle
△
A
B
C
\vartriangle ABC
△
A
BC
holds
∠
A
=
4
0
o
\angle A= 40^o
∠
A
=
4
0
o
and
∠
B
=
2
0
o
\angle B = 20^o
∠
B
=
2
0
o
. The point
P
P
P
lies on the line
A
C
AC
A
C
such that
C
C
C
is between
A
A
A
and
P
P
P
and
∣
C
P
∣
=
∣
A
B
∣
−
∣
B
C
∣
| CP | = | AB | - | BC |
∣
CP
∣
=
∣
A
B
∣
−
∣
BC
∣
. Calculate the
∠
C
B
P
\angle CBP
∠
CBP
.