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National and Regional Contests
Austria Contests
Austrian MO Beginners' Competition
2009 Austria Beginners' Competition
2009 Austria Beginners' Competition
Part of
Austrian MO Beginners' Competition
Subcontests
(4)
3
1
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max integer than can not be represented by stamps 134,..., 143
There are any number of stamps with the values
134
134
134
,
135
135
135
,
.
.
.
...
...
,
142
142
142
and
143
143
143
cents available. Find the largest integer value (in cents) that cannot be represented by these stamps.(G. Woeginger, TU Eindhoven, The Netherlands)[hide=original wording]Es stehen beliebig viele Briefmarken mit den Werten 134, 135. . .., 142 und 143 Cent zur Verfügung. Man bestimme den größten ganzzahligen Wert (in Cent), der nicht durch diese Briefmarken dargestellt werden kann.
4
1
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midpoint wanted, reflect center of square wrt vertex
The center
M
M
M
of the square
A
B
C
D
ABCD
A
BC
D
is reflected wrt
C
C
C
. This gives point
E
E
E
. The intersection of the circumcircle of the triangle
B
D
E
BDE
B
D
E
with the line
A
M
AM
A
M
is denoted by
S
S
S
. Show that
S
S
S
bisects the distance
A
M
AM
A
M
.(W. Janous, WRG Ursulinen, Innsbruck)
2
1
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(x +y^3) (x^3 +y) >= 4x^2y^2 for x,y>=0
Let
x
x
x
and
y
y
y
be nonnegative real numbers. Prove that
(
x
+
y
3
)
(
x
3
+
y
)
≥
4
x
2
y
2
(x +y^3) (x^3 +y) \ge 4x^2y^2
(
x
+
y
3
)
(
x
3
+
y
)
≥
4
x
2
y
2
. When does equality holds?(Task committee)
1
1
Hide problems
positive integers on sides of a square
A positive integer number is written in red on each side of a square. The product of the two red numbers on the adjacent sides is written in green for each corner point. The sum of the green numbers is
40
40
40
. Which values are possible for the sum of the red numbers?(G. Kirchner, University of Innsbruck)