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Germany Team Selection Test
2006 Germany Team Selection Test
2006 Germany Team Selection Test
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Boxes without principle
In a room, there are
2005
2005
2005
boxes, each of them containing one or several sorts of fruits, and of course an integer amount of each fruit. a) Show that we can find
669
669
669
boxes, which altogether contain at least a third of all apples and at least a third of all bananas. b) Can we always find
669
669
669
boxes, which altogether contain at least a third of all apples, at least a third of all bananas and at least a third of all pears?
MOP [Points on the sides of the orthic triangle]
Let
A
1
A_{1}
A
1
,
B
1
B_{1}
B
1
,
C
1
C_{1}
C
1
be the feet of the altitudes of an acute-angled triangle
A
B
C
ABC
A
BC
issuing from the vertices
A
A
A
,
B
B
B
,
C
C
C
, respectively. Let
K
K
K
and
M
M
M
be points on the segments
A
1
C
1
A_{1}C_{1}
A
1
C
1
and
B
1
C
1
B_{1}C_{1}
B
1
C
1
, respectively, such that
∡
K
A
M
=
∡
A
1
A
C
\measuredangle KAM = \measuredangle A_{1}AC
∡
K
A
M
=
∡
A
1
A
C
. Prove that the line
A
K
AK
A
K
is the angle bisector of the angle
C
1
K
M
C_{1}KM
C
1
K
M
.
Geo Ineq! [altitudes integers, inradius prime --> sides = ?]
The lengths of the altitudes of a triangle are positive integers, and the length of the radius of the incircle is a prime number. Find the lengths of the sides of the triangle.