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Problems
Contests
National and Regional Contests
Belgium Contests
Flanders Math Olympiad
1997 Flanders Math Olympiad
1997 Flanders Math Olympiad
Part of
Flanders Math Olympiad
Subcontests
(4)
3
1
Hide problems
area inequality
Δ
o
a
1
b
1
\Delta oa_1b_1
Δ
o
a
1
b
1
is isosceles with
∠
a
1
o
b
1
=
3
6
∘
\angle a_1ob_1 = 36^\circ
∠
a
1
o
b
1
=
3
6
∘
. Construct
a
2
,
b
2
,
a
3
,
b
3
,
.
.
.
a_2,b_2,a_3,b_3,...
a
2
,
b
2
,
a
3
,
b
3
,
...
as below, with
∣
o
a
i
+
1
∣
=
∣
a
i
b
i
∣
|oa_{i+1}| = |a_ib_i|
∣
o
a
i
+
1
∣
=
∣
a
i
b
i
∣
and
∠
a
i
o
b
i
=
3
6
∘
\angle a_iob_i = 36^\circ
∠
a
i
o
b
i
=
3
6
∘
, Call the summed area of the first
k
k
k
triangles
A
k
A_k
A
k
. Let
S
S
S
be the area of the isocseles triangle, drawn in - - -, with top angle
10
8
∘
108^\circ
10
8
∘
and
∣
o
c
∣
=
∣
o
d
∣
=
∣
o
a
1
∣
|oc|=|od|=|oa_1|
∣
oc
∣
=
∣
o
d
∣
=
∣
o
a
1
∣
, going through the points
b
2
b_2
b
2
and
a
2
a_2
a
2
as shown on the picture. (yes,
c
d
cd
c
d
is parallel to
a
1
b
1
a_1b_1
a
1
b
1
there) Show
A
k
<
S
A_k < S
A
k
<
S
for every positive integer
k
k
k
. http://www.mathlinks.ro/Forum/album_pic.php?pic_id=284
2
1
Hide problems
analytical geometry
In the cartesian plane, consider the curves
x
2
+
y
2
=
r
2
x^2+y^2=r^2
x
2
+
y
2
=
r
2
and
(
x
y
)
2
=
1
(xy)^2=1
(
x
y
)
2
=
1
. Call
F
r
F_r
F
r
the convex polygon with vertices the points of intersection of these 2 curves. (if they exist) (a) Find the area of the polygon as a function of
r
r
r
. (b) For which values of
r
r
r
do we have a regular polygon?
1
1
Hide problems
1997 of course ^^
Write the number 1997 as the sum of positive integers for which the product is maximal, and prove there's no better solution.
4
1
Hide problems
Flanders 6 ('97)
Thirteen birds arrive and sit down in a plane. It's known that from each 5-tuple of birds, at least four birds sit on a circle. Determine the greatest
M
∈
{
1
,
2
,
.
.
.
,
13
}
M \in \{1, 2, ..., 13\}
M
∈
{
1
,
2
,
...
,
13
}
such that from these 13 birds, at least
M
M
M
birds sit on a circle, but not necessarily
M
+
1
M + 1
M
+
1
birds sit on a circle. (prove that your
M
M
M
is optimal)