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Contests
National and Regional Contests
Finland Contests
Finnish National High School Mathematics Competition
2008 Finnish National High School Mathematics Competition
2008 Finnish National High School Mathematics Competition
Part of
Finnish National High School Mathematics Competition
Subcontests
(5)
5
1
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Covering a line segment with line segments
The closed line segment
I
I
I
is covered by finitely many closed line segments. Show that one can choose a subfamily
S
S
S
of the family of line segments having the properties: (1) the chosen line segments are disjoint, (2) the sum of the lengths of the line segments of S is more than half of the length of
I
.
I.
I
.
Show that the claim does not hold any more if the line segment
I
I
I
is replaced by a circle and other occurences of the compound word ''line segment" by the word ''circular arc".
4
1
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Maximum number of matches
Eight football teams play matches against each other in such a way that no two teams meet twice and no three teams play all of the three possible matches. What is the largest possible number of matches?
3
1
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Diophantine equation
Solve the diophantine equation
x
2008
−
y
2008
=
2
2009
.
x^{2008}- y^{2008} = 2^{2009}.
x
2008
−
y
2008
=
2
2009
.
2
1
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Perpendicular segments
The incentre of the triangle
A
B
C
ABC
A
BC
is
I
.
I.
I
.
The lines
A
I
,
B
I
AI, BI
A
I
,
B
I
and
C
I
CI
C
I
meet the circumcircle of the triangle
A
B
C
ABC
A
BC
also at points
D
,
E
D, E
D
,
E
and
F
,
F,
F
,
respectively. Prove that
A
D
AD
A
D
and
E
F
EF
EF
are perpendicular.
1
1
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Animals on a rabbit hunt
Foxes, wolves and bears arranged a big rabbit hunt. There were
45
45
45
hunters catching
2008
2008
2008
rabbits. Every fox caught
59
59
59
rabbits, every wolf
41
41
41
rabbits and every bear
40
40
40
rabbits. How many foxes, wolves and bears were there in the hunting company?