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Problems
Contests
National and Regional Contests
Austria Contests
Austrian MO Beginners' Competition
2011 Austria Beginners' Competition
2011 Austria Beginners' Competition
Part of
Austrian MO Beginners' Competition
Subcontests
(4)
4
1
Hide problems
AE=BF wanted, projections of point on circumcircle of isosceles ABC
Let
A
B
C
ABC
A
BC
be an isosceles triangle with
A
C
=
B
C
AC = BC
A
C
=
BC
. On the arc
C
A
CA
C
A
of its circumcircle, which does not contain
B
B
B
, there is a point
P
P
P
. The projection of
C
C
C
on the line
A
P
AP
A
P
is denoted by
E
E
E
, the projection of
C
C
C
on the line
B
P
BP
BP
is denoted by
F
F
F
. Prove that the lines
A
E
AE
A
E
and
B
F
BF
BF
have equal lengths.(W. Janous, WRG Ursulincn, Innsbruck)
3
1
Hide problems
x + y>=2 if x, y>0 with x + y + xy= 3 Austria Beginners' 2011 p3
Let
x
,
y
x, y
x
,
y
be positive real numbers with
x
+
y
+
x
y
=
3
x + y + xy= 3
x
+
y
+
x
y
=
3
. Prove that
x
+
y
≥
2.
x + y\ge 2.
x
+
y
≥
2.
When does equality holds?(K. Czakler, GRG 21, Vienna)
2
1
Hide problems
4 integers wanted in a trinomial
Let
p
p
p
and
q
q
q
be real numbers. The quadratic equation
x
2
+
p
x
+
q
=
0
x^2 + px + q = 0
x
2
+
p
x
+
q
=
0
has the real solutions
x
1
x_1
x
1
and
x
2
x_2
x
2
. In addition, the following two conditions apply: (i) The numbers
x
1
x_1
x
1
and
x
2
x_2
x
2
differ from each other by exactly
1
1
1
. (ii) The numbers
p
p
p
and
q
q
q
differ from each other by exactly
1
1
1
. Show that then
p
p
p
,
q
q
q
,
x
1
x_1
x
1
and
x
2
x_2
x
2
are integers.(G. Kirchner, University of Innsbruck)
1
1
Hide problems
min pos. integer x such that 2x=a^2, 2x=b^3, 5x=c^2
Let
x
x
x
be the smallest positive integer for which
2
x
2x
2
x
is the square of an integer,
3
x
3x
3
x
is the third power of an integer, and
5
x
5x
5
x
is the fifth power of an integer. Find the prime factorization of
x
x
x
.(St. Wagner, Stellenbosch University)