MathDB
Problems
Contests
National and Regional Contests
Belgium Contests
Flanders Math Olympiad
2007 Flanders Math Olympiad
2007 Flanders Math Olympiad
Part of
Flanders Math Olympiad
Subcontests
(4)
4
1
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Functional equation
If
f
,
g
:
R
→
R
f,g: \mathbb{R} \to \mathbb{R}
f
,
g
:
R
→
R
are functions that satisfy
f
(
x
+
g
(
y
)
)
=
2
x
+
y
f(x+g(y)) = 2x+y
f
(
x
+
g
(
y
))
=
2
x
+
y
∀
x
,
y
∈
R
\forall x,y \in \mathbb{R}
∀
x
,
y
∈
R
, then determine
g
(
x
+
f
(
y
)
)
g(x+f(y))
g
(
x
+
f
(
y
))
.
3
1
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flemish areas
Let
A
B
C
D
ABCD
A
BC
D
be a square with side
10
10
10
. Let
M
M
M
and
N
N
N
be the midpoints of
[
A
B
]
[AB]
[
A
B
]
and
[
B
C
]
[BC]
[
BC
]
respectively. Three circles are drawn: one with midpoint
D
D
D
and radius
∣
A
D
∣
|AD|
∣
A
D
∣
, one with midpoint
M
M
M
and radius
∣
A
M
∣
|AM|
∣
A
M
∣
, and one with midpoint
N
N
N
and radius
∣
B
N
∣
|BN|
∣
BN
∣
. The three circles intersect in the points
R
,
S
R, S
R
,
S
and
T
T
T
inside the square. Determine the area of
△
R
S
T
\triangle RST
△
RST
.
2
1
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easy geometry
Given is a half circle with midpoint
O
O
O
and diameter
A
B
AB
A
B
. Let
Z
Z
Z
be a random point inside the half circle, and let
X
X
X
be the intersection of
O
Z
OZ
OZ
and the half circle, and
Y
Y
Y
the intersection of
A
Z
AZ
A
Z
and the half circle.If
P
P
P
is the intersection of
B
Y
BY
B
Y
with the tangent line in
X
X
X
to the half circle, show that
P
Z
⊥
B
X
PZ \perp BX
PZ
⊥
BX
.
1
1
Hide problems
easy problem
1. The numbers
1
,
2
,
…
1,2, \ldots
1
,
2
,
…
are placed in a triangle as following:
1
2
3
4
5
6
7
8
9
10
…
\begin{matrix} 1 & & & \\ 2 & 3 & & \\ 4 & 5 & 6 & \\ 7 & 8 & 9 & 10 \\ \ldots \end{matrix}
1
2
4
7
…
3
5
8
6
9
10
What is the sum of the numbers on the
n
n
n
-th row?