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Austrian MO Beginners' Competition
2005 Austria Beginners' Competition
3
3
Part of
2005 Austria Beginners' Competition
Problems
(1)
[ x ] + {y } =z, [ y ] + {z } =x, [ z ] + {x } =y
Source: 2005 Austria Beginners' Competition p3
10/12/2022
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
floor function
system of equations
System
algebra