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National and Regional Contests
Austria Contests
Austrian MO Beginners' Competition
2002 Austria Beginners' Competition
2002 Austria Beginners' Competition
Part of
Austrian MO Beginners' Competition
Subcontests
(4)
1
1
Hide problems
product of sum of seven consecutive pairs
We calculate the sum of
7
7
7
natural consecutive pairs (e.g.
2
+
4
+
6
+
8
+
10
+
12
+
14
2+4+6+8+10+12+14
2
+
4
+
6
+
8
+
10
+
12
+
14
) and we will call the result
A
A
A
, then the sum of the next
7
7
7
consecutive pairs (in the example,
16
+
18
+
.
.
.
16+ 18+...
16
+
18
+
...
) and its result we will call
B
B
B
, and finally we calculate the sum of the following
7
7
7
consecutive pairs and its result we will call
C
C
C
. Can the product
A
B
C
ABC
A
BC
be
200
2
3
2002^3
200
2
3
?
2
1
Hide problems
x^[x]=9/2
Prove that there are no
x
∈
R
+
x\in\mathbb{R}^+
x
∈
R
+
such that
x
⌊
x
⌋
=
9
2
.
x^{\lfloor x \rfloor }=\frac92.
x
⌊
x
⌋
=
2
9
.
3
1
Hide problems
|x^2-4x+1|>|x^2-4x+5|
Find all real numbers
x
x
x
that satisfy the following inequality
∣
x
2
−
4
x
+
1
∣
>
∣
x
2
−
4
x
+
5
∣
|x^2-4x+1|>|x^2-4x+5|
∣
x
2
−
4
x
+
1∣
>
∣
x
2
−
4
x
+
5∣
4
1
Hide problems
construction of trapezoid and area wanted, <120^o
In a trapezoid
A
B
C
D
ABCD
A
BC
D
with base
A
B
AB
A
B
let
E
E
E
be the midpoint of side
A
D
AD
A
D
. Suppose further that
2
C
D
=
E
C
=
B
C
=
b
2CD=EC=BC=b
2
C
D
=
EC
=
BC
=
b
. Let
∠
E
C
B
=
12
0
∘
\angle ECB=120^{\circ}
∠
ECB
=
12
0
∘
. Construct the trapezoid and determine its area based on
b
b
b
.