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Problems
Contests
National and Regional Contests
Belgium Contests
Flanders Math Olympiad
1993 Flanders Math Olympiad
1993 Flanders Math Olympiad
Part of
Flanders Math Olympiad
Subcontests
(4)
2
1
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jeweler
A jeweler covers the diagonal of a unit square with small golden squares in the following way: - the sides of all squares are parallel to the sides of the unit square - for each neighbour is their sidelength either half or double of that square (squares are neighbour if they share a vertex) - each midpoint of a square has distance to the vertex of the unit square equal to
1
2
,
1
4
,
1
8
,
.
.
.
\dfrac12, \dfrac14, \dfrac18, ...
2
1
,
4
1
,
8
1
,
...
of the diagonal. (so real length:
×
2
\times \sqrt2
×
2
) - all midpoints are on the diagonal (a) What is the side length of the middle square? (b) What is the total gold-plated area? http://www.mathlinks.ro/Forum/album_pic.php?pic_id=281
4
1
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trig product
Define the sequence
o
a
n
oa_n
o
a
n
as follows:
o
a
0
=
1
,
o
a
n
=
o
a
n
−
1
⋅
c
o
s
(
π
2
n
+
1
)
oa_0=1, oa_n= oa_{n-1} \cdot cos\left( \dfrac{\pi}{2^{n+1}} \right)
o
a
0
=
1
,
o
a
n
=
o
a
n
−
1
⋅
cos
(
2
n
+
1
π
)
. Find
lim
n
→
+
∞
o
a
n
\lim\limits_{n\rightarrow+\infty} oa_n
n
→
+
∞
lim
o
a
n
.
3
1
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inequality (maybe posted before?)
For
a
,
b
,
c
>
0
a,b,c>0
a
,
b
,
c
>
0
we have:
−
1
<
(
a
−
b
a
+
b
)
1993
+
(
b
−
c
b
+
c
)
1993
+
(
c
−
a
c
+
a
)
1993
<
1
-1 < \left(\dfrac{a-b}{a+b}\right)^{1993} + \left(\dfrac{b-c}{b+c}\right)^{1993} + \left(\dfrac{c-a}{c+a}\right)^{1993} < 1
−
1
<
(
a
+
b
a
−
b
)
1993
+
(
b
+
c
b
−
c
)
1993
+
(
c
+
a
c
−
a
)
1993
<
1
1
1
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easy combinatorics
The 20 pupils in a class each send 10 cards to 10 (different) class members. [note: you cannot send a card to yourself.] (a) Show at least 2 pupils sent each other a card. (b) Now suppose we had
n
n
n
pupils sending
m
m
m
cards each. For which
(
m
,
n
)
(m,n)
(
m
,
n
)
is the above true? (That is, find minimal
m
(
n
)
m(n)
m
(
n
)
or maximal
n
(
m
)
n(m)
n
(
m
)
)