MathDB
Problems
Contests
National and Regional Contests
Belgium Contests
Flanders Math Olympiad
1987 Flanders Math Olympiad
1987 Flanders Math Olympiad
Part of
Flanders Math Olympiad
Subcontests
(4)
1
1
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m x n grid in a rectangle, no of rectangles wanted
A rectangle
A
B
C
D
ABCD
A
BC
D
is given. On the side
A
B
AB
A
B
,
n
n
n
different points are chosen strictly between
A
A
A
and
B
B
B
. Similarly,
m
m
m
different points are chosen on the side
A
D
AD
A
D
. Lines are drawn from the points parallel to the sides. How many rectangles are formed in this way? (One possibility is shown in the figure.) https://cdn.artofproblemsolving.com/attachments/0/1/dcf48e4ce318fdcb8c7088a34fac226e26e246.png
2
1
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2 XY <= AB+ A'B" , midpoints inequality by 3 pairs of parallel lines
Two parallel lines
a
a
a
and
b
b
b
meet two other lines
c
c
c
and
d
d
d
. Let
A
A
A
and
A
′
A'
A
′
be the points of intersection of
a
a
a
with
c
c
c
and
d
d
d
, respectively. Let
B
B
B
and
B
′
B'
B
′
be the points of intersection of
b
b
b
with
c
c
c
and
d
d
d
, respectively. If
X
X
X
is the midpoint of the line segment
A
A
′
A A'
A
A
′
and
Y
Y
Y
is the midpoint of the segment
B
B
′
BB'
B
B
′
, prove that
∣
X
Y
∣
≤
∣
A
B
∣
+
∣
A
′
B
′
∣
2
.
|XY| \le \frac{|AB|+|A'B'|}{2}.
∣
X
Y
∣
≤
2
∣
A
B
∣
+
∣
A
′
B
′
∣
.
4
1
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A nice limit
Show that for
p
>
1
p>1
p
>
1
we have
lim
n
→
+
∞
1
p
+
2
p
+
.
.
.
+
(
n
−
1
)
p
+
n
p
+
(
n
−
1
)
p
+
.
.
.
+
2
p
+
1
p
n
2
=
+
∞
\lim_{n\rightarrow+\infty}\frac{1^p+2^p+...+(n-1)^p+n^p+(n-1)^p+...+2^p+1^p}{n^2} = +\infty
n
→
+
∞
lim
n
2
1
p
+
2
p
+
...
+
(
n
−
1
)
p
+
n
p
+
(
n
−
1
)
p
+
...
+
2
p
+
1
p
=
+
∞
Find the limit if
p
=
1
p=1
p
=
1
.
3
1
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[to be solved] - functional equation
Find all continuous functions
f
:
R
→
R
f: \mathbb{R}\rightarrow\mathbb{R}
f
:
R
→
R
such that
f
(
x
)
3
=
−
x
12
⋅
(
x
2
+
7
x
⋅
f
(
x
)
+
16
⋅
f
(
x
)
2
)
,
∀
x
∈
R
.
f(x)^3 = -\frac x{12}\cdot\left(x^2+7x\cdot f(x)+16\cdot f(x)^2\right),\ \forall x \in \mathbb{R}.
f
(
x
)
3
=
−
12
x
⋅
(
x
2
+
7
x
⋅
f
(
x
)
+
16
⋅
f
(
x
)
2
)
,
∀
x
∈
R
.