MathDB
Problems
Contests
National and Regional Contests
Belgium Contests
Flanders Math Olympiad
2020 Flanders Math Olympiad
2020 Flanders Math Olympiad
Part of
Flanders Math Olympiad
Subcontests
(4)
4
1
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n hoops on a circle
There are
n
n
n
hoops on a circle. Rik numbers all hoops with a natural number so that all numbers from
1
1
1
to
n
n
n
occur exactly once. Then he makes one walk from hoop to hoop. He starts in hoop
1
1
1
and then follows the following rule: if he gets to hoop
k
k
k
, then he walks to the hoop that places
k
k
k
clockwise without getting into the intermediate hoops. The walk ends when Rik has to walk to a hoop he has already been to. The length of the walk is the number of hoops he passed on the way. For example, for
n
=
6
n = 6
n
=
6
Rik can take a walk of length
5
5
5
as the hoops are numbered as shown in the figure. https://cdn.artofproblemsolving.com/attachments/2/a/3d4b7edbba4d145c7e00368f9b794f39572dc5.png (a) Determine for every even
n
n
n
how Rik can number the hoops so that he has one walk of length
n
n
n
.(b) Determine for every odd
n
n
n
how Rik can number the hoops so that he has one walk of length
n
−
1
n - 1
n
−
1
.(c) Show that for an odd
n
n
n
there is no such numbering of the hoops that Rik can make a walk of length
n
n
n
.
2
1
Hide problems
ISBN code combiatorics
Every officially published book used to have an ISBN code (International Standard Book Number) which consisted of
10
10
10
symbols. Such code looked like this:
a
1
a
2
.
.
.
a
9
a
10
a_1a_2 . . . a_9a_{10}
a
1
a
2
...
a
9
a
10
with
a
1
,
.
.
.
,
a
9
∈
{
0
,
1
,
.
.
.
,
9
}
a_1, . . . , a_9 \in \{0, 1, . . . , 9\}
a
1
,
...
,
a
9
∈
{
0
,
1
,
...
,
9
}
and
a
10
∈
{
0
,
1
,
.
.
.
,
9
,
X
}
a_{10} \in \{0, 1, . . . , 9, X\}
a
10
∈
{
0
,
1
,
...
,
9
,
X
}
. The symbol
X
X
X
stood for the number
10
10
10
. With a valid ISBN code was
a
1
+
2
a
2
+
.
.
.
+
9
a
9
+
10
a
10
a_1 + 2a2 + . . . + 9a_9 + 10a_{10}
a
1
+
2
a
2
+
...
+
9
a
9
+
10
a
10
a multiple of
11
11
11
. Prove the following statements. (a) If one symbol is changed in a valid ISBN code, the result is no valid ISBN code. (b) When two different symbols swap places in a valid ISBN code then the result is not a valid ISBN.
1
1
Hide problems
1/9 sin^4 x + 1/16 cos^4 x = 1/25
Let
x
x
x
be an angle between
0
o
0^o
0
o
and
9
0
o
90^o
9
0
o
so that
sin
4
x
9
+
cos
4
x
16
=
1
25
.
\frac{\sin^4 x}{9}+\frac{\cos^4 x}{16 }=\frac{1}{25} .
9
sin
4
x
+
16
cos
4
x
=
25
1
.
Then what is
tan
x
\tan x
tan
x
?
3
1
Hide problems
angle bisector wanted inside a regular pentagon
The point
M
M
M
is the center of a regular pentagon
A
B
C
D
E
ABCDE
A
BC
D
E
. The point
P
P
P
is an inner point of the line segment
[
D
M
]
[DM]
[
D
M
]
. The circumscribed circle of triangle
△
A
B
P
\vartriangle ABP
△
A
BP
intersects the side
[
A
E
]
[AE]
[
A
E
]
at point
Q
Q
Q
(different from
A
A
A
). The perpendicular from
P
P
P
on
C
D
CD
C
D
intersects the side
[
A
E
]
[AE]
[
A
E
]
at point
S
S
S
. Prove that
P
S
PS
PS
is the bisector of
∠
A
P
Q
\angle APQ
∠
A
PQ
.