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2
Fractionary parts of cubes in Q/Z
Fractionary parts of cubes in Q/Z
Source: Romania,2014,First TST,Problem 2
July 18, 2014
induction
modular arithmetic
number theory
algebra
Problem Statement
Let
n
≥
2
n \ge 2
n
≥
2
be an integer. Show that there exist
n
+
1
n+1
n
+
1
numbers
x
1
,
x
2
,
…
,
x
n
+
1
∈
Q
∖
Z
x_1, x_2, \ldots, x_{n+1} \in \mathbb{Q} \setminus \mathbb{Z}
x
1
,
x
2
,
…
,
x
n
+
1
∈
Q
∖
Z
, so that
{
x
1
3
}
+
{
x
2
3
}
+
⋯
+
{
x
n
3
}
=
{
x
n
+
1
3
}
\{ x_1^3 \} + \{ x_2^3 \} + \cdots + \{ x_n^3 \}=\{ x_{n+1}^3 \}
{
x
1
3
}
+
{
x
2
3
}
+
⋯
+
{
x
n
3
}
=
{
x
n
+
1
3
}
, where
{
x
}
\{ x \}
{
x
}
is the fractionary part of
x
x
x
.
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