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National and Regional Contests
Austria Contests
Austrian MO Beginners' Competition
2010 Austria Beginners' Competition
2010 Austria Beginners' Competition
Part of
Austrian MO Beginners' Competition
Subcontests
(3)
4
1
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angles wanted in right triangle, BC<AC, DE=AB, perp. bisector
In the right-angled triangle
A
B
C
ABC
A
BC
with a right angle at
C
C
C
, the side
B
C
BC
BC
is longer than the side
A
C
AC
A
C
. The perpendicular bisector of
A
B
AB
A
B
intersects the line
B
C
BC
BC
at point
D
D
D
and the line
A
C
AC
A
C
at point
E
E
E
. The segments
D
E
DE
D
E
has the same length as the side
A
B
AB
A
B
. Find the measures of the angles of the triangle
A
B
C
ABC
A
BC
.(R. Henner, Vienna)
2
1
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tree's problem with average Austria Beginners' Competition 2010 p2
In a national park there is a group of sequoia trees, all of which have a positive integer age. Their average age is
41
41
41
years. After a
2010
2010
2010
year old building was destroyed by lightning, the average age drops to
40
40
40
years. How many trees were originally in the group? At most, how many of them were exactly
2010
2010
2010
years old?(W. Janous, WRG Ursulinen, Innsbruck)
1
1
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2010 not a^2-b^2 Austria Beginners' Competition 2010 p1
Prove that
2010
2010
2010
cannot be represented as the difference between two square numbers.(B. Schmidt, Graz University of Technology)