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Contests
National and Regional Contests
Belgium Contests
Flanders Math Olympiad
2011 Flanders Math Olympiad
2011 Flanders Math Olympiad
Part of
Flanders Math Olympiad
Subcontests
(4)
1
1
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cx^2 - bx + a >cx - b if ax^2+bx+c >cx
Given are three numbers
a
,
b
,
c
∈
R
−
{
0
}
a, b, c \in R-\{0\}
a
,
b
,
c
∈
R
−
{
0
}
. The parabola with equation
y
=
a
x
2
+
b
x
+
c
y = ax^2+bx+c
y
=
a
x
2
+
b
x
+
c
lies above the line
y
=
c
x
y = cx
y
=
c
x
. Prove that the parabola with equation
y
=
c
x
2
−
b
x
+
a
y = cx^2 - bx + a
y
=
c
x
2
−
b
x
+
a
lies above the line
y
=
c
x
−
b
y = cx - b
y
=
c
x
−
b
.
3
1
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18 students in a class
There are
18
18
18
students in a class. Each student is asked two questions: how many other students have the same first name as you and how many other students have the same surname as you. The answers
0
,
1
,
2
,
.
.
.
,
7
0, 1, 2, . . ., 7
0
,
1
,
2
,
...
,
7
all occur. Prove that there are two students with the same first name and last name.
2
1
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ratio of volumes, truncated cone inscribed in sphere (2011 VWO Flanders MO p2)
The area of the ground plane of a truncated cone
K
K
K
is four times as large as the surface of the top surface. A sphere
B
B
B
is circumscribed in
K
K
K
, that is to say that
B
B
B
touches both the top surface and the base and the sides. Calculate ratio volume
B
:
B :
B
:
Volume
K
K
K
.
4
1
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ratio chasing with 2 intersecting cevians
Given is a triangle
A
B
C
ABC
A
BC
and points
D
D
D
and
E
E
E
, respectively on
]
B
C
[
] BC [
]
BC
[
and
]
A
B
[
] AB [
]
A
B
[
.
F
F
F
it is intersection of lines
A
D
AD
A
D
and
C
E
CE
CE
. We denote as
∣
C
D
∣
=
a
,
∣
B
D
∣
=
b
,
∣
D
F
∣
=
c
| CD | = a, | BD | = b, | DF | = c
∣
C
D
∣
=
a
,
∣
B
D
∣
=
b
,
∣
D
F
∣
=
c
and
∣
A
F
∣
=
d
| AF | = d
∣
A
F
∣
=
d
. Determine the ratio
∣
B
E
∣
∣
A
E
∣
\frac{| BE |}{|AE |}
∣
A
E
∣
∣
BE
∣
in terms of
a
,
b
,
c
a, b, c
a
,
b
,
c
and
d
d
d
https://cdn.artofproblemsolving.com/attachments/5/7/856c97045db2d9a26841ad00996a2b809addaa.png