MathDB
Problems
Contests
National and Regional Contests
Austria Contests
Austrian MO Beginners' Competition
2012 Austria Beginners' Competition
2012 Austria Beginners' Competition
Part of
Austrian MO Beginners' Competition
Subcontests
(4)
4
1
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DE and DF trisect AB, equilaterals ABC and ADB
A segment
A
B
AB
A
B
is given. We erect the equilateral triangles
A
B
C
ABC
A
BC
and
A
D
B
ADB
A
D
B
above and below
A
B
AB
A
B
. We denote the midpoints of
A
C
AC
A
C
and
B
C
BC
BC
by
E
E
E
and
F
F
F
respectively. Prove that the straight lines
D
E
DE
D
E
and
D
F
DF
D
F
divide the segment
A
B
AB
A
B
into three parts of equal length .
3
1
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4ab <= 2 (a^2+ b^2) <= 5 ab if 0< a <= 2b <= 4a
Let
a
a
a
and
b
b
b
be two positive real numbers with
a
≤
2
b
≤
4
a
a \le 2b \le 4a
a
≤
2
b
≤
4
a
. Prove that
4
a
b
≤
2
(
a
2
+
b
2
)
≤
5
a
b
4ab \le2 (a^2+ b^2) \le 5 ab
4
ab
≤
2
(
a
2
+
b
2
)
≤
5
ab
.
2
1
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divide n packages with weights 1, 2, 3, 4, n into 3 groups of equal weight
A postman wants to divide
n
n
n
packages with weights
1
,
2
,
3
,
4
,
n
1, 2, 3, 4, n
1
,
2
,
3
,
4
,
n
into three groups of exactly the same weight. Can he do this if (a)
n
=
2011
n = 2011
n
=
2011
? (b)
n
=
2012
n = 2012
n
=
2012
?
1
1
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14 divides ab if 7a + 8b = 14c + 28d
Let
a
,
b
,
c
a, b, c
a
,
b
,
c
and
d
d
d
be four integers such that
7
a
+
8
b
=
14
c
+
28
d
7a + 8b = 14c + 28d
7
a
+
8
b
=
14
c
+
28
d
. Prove that the product
a
⋅
b
a\cdot b
a
⋅
b
is always divisible by
14
14
14
.