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Problems
Contests
National and Regional Contests
Belgium Contests
Flanders Math Olympiad
2012 Flanders Math Olympiad
2012 Flanders Math Olympiad
Part of
Flanders Math Olympiad
Subcontests
(4)
3
1
Hide problems
| sin m \theta_2 - sin m \theta_1| <= m| sin (\theta_2 - \theta_1)|
(a) Show that for any angle
θ
\theta
θ
and for any natural number
m
m
m
:
∣
sin
m
θ
∣
≤
m
∣
sin
θ
∣
| \sin m\theta| \le m| \sin \theta|
∣
sin
m
θ
∣
≤
m
∣
sin
θ
∣
(b) Show that for all angles
θ
1
\theta_1
θ
1
and
θ
2
\theta_2
θ
2
and for all even natural numbers
m
m
m
:
∣
sin
m
θ
2
−
sin
m
θ
1
∣
≤
m
∣
sin
(
θ
2
−
θ
1
)
∣
| \sin m \theta_2 - \sin m \theta_1| \le m| \sin (\theta_2 - \theta_1)|
∣
sin
m
θ
2
−
sin
m
θ
1
∣
≤
m
∣
sin
(
θ
2
−
θ
1
)
∣
(c) Show that for every odd natural number
m
m
m
there are two angles, resp.
θ
1
\theta_1
θ
1
and
θ
2
\theta_2
θ
2
, exist for which the inequality in (b) is not valid.
1
1
Hide problems
alpha - beta - gamma tournament in a class
Our class decides to have a alpha - beta - gamma tournament. This party game is always played in groups of three. Any possible combination of three players (three students or two students and the teacher) plays the game
1
1
1
time. The player who wins gets
1
1
1
point. The two losers get no points. At the end of the tournament, miraculously, all students have as many points. The teacher has
3
3
3
points. How many students are there in our class?
2
1
Hide problems
n - ab is a perfect square
Let
n
n
n
be a natural number. Call
a
a
a
the smallest natural number you need to subtract from
n
n
n
to get a perfect square. Call
b
b
b
the smallest natural number that you must add to
n
n
n
to get a perfect square. Prove that
n
−
a
b
n - ab
n
−
ab
is a perfect square.
4
1
Hide problems
angle chasing when <A=66^o, BD= AB + AC , external angle bisctor of A
In
△
A
B
C
,
∠
A
=
6
6
o
\vartriangle ABC, \angle A = 66^o
△
A
BC
,
∠
A
=
6
6
o
and
∣
A
B
∣
<
∣
A
C
∣
| AB | <| AC |
∣
A
B
∣
<
∣
A
C
∣
. The outer bisector in
A
A
A
intersects
B
C
BC
BC
in
D
D
D
and
∣
B
D
∣
=
∣
A
B
∣
+
∣
A
C
∣
| BD | = | AB | + | AC |
∣
B
D
∣
=
∣
A
B
∣
+
∣
A
C
∣
. Determine the angles of
△
A
B
C
\vartriangle ABC
△
A
BC
.