MathDB
Problems
Contests
International Contests
Pan African
2017 Pan African
Problem 4
Problem 4
Part of
2017 Pan African
Problems
(1)
Problem 4 Equation with integer part
Source: PAMO 2017 Problem 4
7/5/2017
Find all the real numbers
x
x
x
such that
1
[
x
]
+
1
[
2
x
]
=
{
x
}
+
1
3
\frac{1}{[x]}+\frac{1}{[2x]}=\{x\}+\frac{1}{3}
[
x
]
1
+
[
2
x
]
1
=
{
x
}
+
3
1
where
[
x
]
[x]
[
x
]
denotes the integer part of
x
x
x
and
{
x
}
=
x
−
[
x
]
\{x\}=x-[x]
{
x
}
=
x
−
[
x
]
. For example,
[
2.5
]
=
2
,
{
2.5
}
=
0.5
[2.5]=2, \{2.5\} = 0.5
[
2.5
]
=
2
,
{
2.5
}
=
0.5
and
[
−
1.7
]
=
−
2
,
{
−
1.7
}
=
0.3
[-1.7]= -2, \{-1.7\} = 0.3
[
−
1.7
]
=
−
2
,
{
−
1.7
}
=
0.3
algebra
Integer Part