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Problems
Contests
National and Regional Contests
Belgium Contests
Flanders Math Olympiad
1998 Flanders Math Olympiad
1998 Flanders Math Olympiad
Part of
Flanders Math Olympiad
Subcontests
(4)
4
1
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biljart
A billiard table. (see picture) A white ball is on
p
1
p_1
p
1
and a red ball is on
p
2
p_2
p
2
. The white ball is shot towards the red ball as shown on the pic, hitting 3 sides first. Find the minimal distance the ball must travel. http://www.mathlinks.ro/Forum/album_pic.php?pic_id=280
2
1
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cube
Given a cube with edges of length 1,
e
e
e
the midpoint of
[
b
c
]
[bc]
[
b
c
]
, and
m
m
m
midpoint of the face
c
d
c
1
d
1
cdc_1d_1
c
d
c
1
d
1
, as on the figure. Find the area of intersection of the cube with the plane through the points
a
,
m
,
e
a,m,e
a
,
m
,
e
. http://www.mathlinks.ro/Forum/album_pic.php?pic_id=279
1
1
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sum of numbers with maxized gcd
Prove there exist positive integers a,b,c for which
a
+
b
+
c
=
1998
a+b+c=1998
a
+
b
+
c
=
1998
, the gcd is maximized, and
0
<
a
<
b
≤
c
<
2
a
0<a<b\leq c<2a
0
<
a
<
b
≤
c
<
2
a
. Find those numbers. Are they unique?
3
1
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Magical squares (flanders '98)
a magical
3
×
3
3\times3
3
×
3
square is a
3
×
3
3\times3
3
×
3
matrix containing all number from 1 to 9, and of which the sum of every row, every column, every diagonal, are all equal. Determine all magical
3
×
3
3\times3
3
×
3
square