MathDB
Problems
Contests
National and Regional Contests
Austria Contests
Austrian MO Beginners' Competition
2005 Austria Beginners' Competition
2005 Austria Beginners' Competition
Part of
Austrian MO Beginners' Competition
Subcontests
(4)
4
1
Hide problems
exists a triangle with side lengths AU, BV, CW and find it's area
We are given the triangle
A
B
C
ABC
A
BC
with an area of
2000
2000
2000
. Let
P
,
Q
,
R
P,Q,R
P
,
Q
,
R
be the midpoints of the sides
B
C
BC
BC
,
A
C
AC
A
C
,
A
B
AB
A
B
. Let
U
,
V
,
W
U,V,W
U
,
V
,
W
be the midpoints of the sides
Q
R
QR
QR
,
P
R
PR
PR
,
P
Q
PQ
PQ
. The lengths of the line segments
A
U
AU
A
U
,
B
V
BV
B
V
,
C
W
CW
C
W
are
x
x
x
,
y
y
y
,
z
z
z
. Show that there exists a triangle with side lengths
x
x
x
,
y
y
y
and
z
z
z
and caluclate it's area.
3
1
Hide problems
[ x ] + {y } =z, [ y ] + {z } =x, [ z ] + {x } =y
Determine all triples
(
x
,
y
,
z
)
(x,y,z)
(
x
,
y
,
z
)
of real numbers that satisfy all of the following three equations:
{
⌊
x
⌋
+
{
y
}
=
z
⌊
y
⌋
+
{
z
}
=
x
⌊
z
⌋
+
{
x
}
=
y
\begin{cases} \lfloor x \rfloor + \{y\} =z \\ \lfloor y \rfloor + \{z\} =x \\ \lfloor z \rfloor + \{x\} =y \end{cases}
⎩
⎨
⎧
⌊
x
⌋
+
{
y
}
=
z
⌊
y
⌋
+
{
z
}
=
x
⌊
z
⌋
+
{
x
}
=
y
2
1
Hide problems
(|x| - 2)^2 + (|y| - 2)^2 < 5 , diophantine
Determine the number of integer pairs
(
x
,
y
)
(x, y)
(
x
,
y
)
such that
(
∣
x
∣
−
2
)
2
+
(
∣
y
∣
−
2
)
2
<
5
(|x| - 2)^2 + (|y| - 2)^2 < 5
(
∣
x
∣
−
2
)
2
+
(
∣
y
∣
−
2
)
2
<
5
.
1
1
Hide problems
4a(a + 1) = b(b + 3) diophantine
Show that there are no positive integers
a
a
a
und
b
b
b
such that
4
a
(
a
+
1
)
=
b
(
b
+
3
)
4a(a + 1) = b(b + 3)
4
a
(
a
+
1
)
=
b
(
b
+
3
)