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Contests
National and Regional Contests
Czech Republic Contests
Czech and Slovak Olympiad III A
2007 Czech and Slovak Olympiad III A
6
6
Part of
2007 Czech and Slovak Olympiad III A
Problems
(1)
find x,y,z for which two sets are equal
Source: Czech and Slovak third round,2007,p6
3/3/2012
Find all pariwise distinct real numbers
x
,
y
,
z
x,y,z
x
,
y
,
z
such that
{
x
−
y
y
−
z
,
y
−
z
z
−
x
,
z
−
x
x
−
y
}
=
{
x
,
y
,
z
}
\left\{\frac{x-y}{y-z},\frac{y-z}{z-x},\frac{z-x}{x-y} \right\} = \{x,y,z\}
{
y
−
z
x
−
y
,
z
−
x
y
−
z
,
x
−
y
z
−
x
}
=
{
x
,
y
,
z
}
. (It means, those three fractions make a permutation of
x
,
y
x, y
x
,
y
, and
z
z
z
.)
algebra unsolved
algebra