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Problems
Contests
National and Regional Contests
Austria Contests
Austrian MO Beginners' Competition
2008 Austria Beginners' Competition
2008 Austria Beginners' Competition
Part of
Austrian MO Beginners' Competition
Subcontests
(4)
1
1
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2^n/ n^2 is integer
Determine all positive integers
n
n
n
such that
2
n
n
2
\frac{2^n}{n^2}
n
2
2
n
is an integer.
3
1
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(a + b)/(a^2 -ab + b^2) <= 4/ |a + b|
Prove the inequality
a
+
b
a
2
−
a
b
+
b
2
≤
4
∣
a
+
b
∣
\frac{a + b}{a^2 -ab + b^2} \le \frac{4}{ |a + b|}
a
2
−
ab
+
b
2
a
+
b
≤
∣
a
+
b
∣
4
for all real numbers
a
a
a
and
b
b
b
with
a
+
b
≠
0
a + b\ne 0
a
+
b
=
0
. When does equality hold?
2
1
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x [x [x]] =\sqrt2
Determine all real numbers
x
x
x
satisfying
x
⌊
x
⌊
x
⌋
⌋
=
2
.
x \lfloor x \lfloor x \rfloor \rfloor =\sqrt2.
x
⌊
x
⌊
x
⌋⌋
=
2
.
4
1
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<BAC=? if bisector of <BAC, B-altitude and perp. bisector of AB concur
Let
A
B
C
ABC
A
BC
be an acute-angled triangle with the property that the bisector of
∠
B
A
C
\angle BAC
∠
B
A
C
, the altitude through
B
B
B
and the perpendicular bisector of
A
B
AB
A
B
intersect in one point. Determine the angle
α
=
∠
B
A
C
\alpha = \angle BAC
α
=
∠
B
A
C
.