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Mediterranean Mathematics Olympiad
2007 Mediterranean Mathematics Olympiad
4
Inequality on integer part and decimal part of x
Inequality on integer part and decimal part of x
Source: Mediterranean MO 2007
October 31, 2010
inequalities
floor function
inequalities unsolved
Problem Statement
Let
x
>
1
x > 1
x
>
1
be a non-integer number. Prove that
(
x
+
{
x
}
[
x
]
−
[
x
]
x
+
{
x
}
)
+
(
x
+
[
x
]
{
x
}
−
{
x
}
x
+
[
x
]
)
>
9
2
\biggl( \frac{x+\{x\}}{[x]} - \frac{[x]}{x+\{x\}} \biggr) + \biggl( \frac{x+[x]}{ \{x \} } - \frac{ \{ x \}}{x+[x]} \biggr) > \frac 92
(
[
x
]
x
+
{
x
}
−
x
+
{
x
}
[
x
]
)
+
(
{
x
}
x
+
[
x
]
−
x
+
[
x
]
{
x
}
)
>
2
9
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