MathDB
Problems
Contests
National and Regional Contests
Taiwan Contests
TST Round 1
2023 Taiwan TST Round 1
6
6
Part of
2023 Taiwan TST Round 1
Problems
(1)
Sum of Decimal Parts
Source: 2023 Taiwan Round 1 Mock Exam P6
3/18/2023
For every positive integer
M
≥
2
M \geq 2
M
≥
2
, find the smallest real number
C
M
C_M
C
M
such that for any integers
a
1
,
a
2
,
…
,
a
2023
a_1, a_2,\ldots , a_{2023}
a
1
,
a
2
,
…
,
a
2023
, there always exist some integer
1
≤
k
<
M
1 \leq k < M
1
≤
k
<
M
such that
{
k
a
1
M
}
+
{
k
a
2
M
}
+
⋯
+
{
k
a
2023
M
}
≤
C
M
.
\left\{\frac{ka_1}{M}\right\}+\left\{\frac{ka_2}{M}\right\}+\cdots+\left\{\frac{ka_{2023}}{M}\right\}\leq C_M.
{
M
k
a
1
}
+
{
M
k
a
2
}
+
⋯
+
{
M
k
a
2023
}
≤
C
M
.
Here,
{
x
}
\{x\}
{
x
}
is the unique number in the interval
[
0
,
1
)
[0, 1)
[
0
,
1
)
such that
x
−
{
x
}
x - \{x\}
x
−
{
x
}
is an integer.Proposed by usjl
Taiwan
algebra