MathDB
Problems
Contests
National and Regional Contests
China Contests
XES Mathematics Olympiad
the 14th XMO
P2
P2
Part of
the 14th XMO
Problems
(1)
Fractional parts
Source: 14th XMO P2
1/14/2024
Let
p
p
p
be a prime. Define
f
n
(
k
)
f_n(k)
f
n
(
k
)
to be the number of positive integers
1
≤
x
≤
p
−
1
1\leq x\leq p-1
1
≤
x
≤
p
−
1
such that
(
{
x
p
}
−
{
k
p
}
)
(
{
n
x
p
}
−
{
k
p
}
)
<
0.
\left(\left\{\frac{x}{p}\right\}-\left\{\frac{k}{p}\right\}\right)\left(\left\{\frac{nx}{p}\right\}-\left\{\frac{k}{p}\right\}\right)<0.
(
{
p
x
}
−
{
p
k
}
)
(
{
p
n
x
}
−
{
p
k
}
)
<
0.
Let
a
n
=
f
n
(
1
2
)
+
f
n
(
3
2
)
+
⋯
+
f
n
(
2
p
−
1
2
)
a_n=f_n\left(\frac 12\right)+f_n\left(\frac 32\right)+\dots+f_n\left(\frac{2p-1}{2}\right)
a
n
=
f
n
(
2
1
)
+
f
n
(
2
3
)
+
⋯
+
f
n
(
2
2
p
−
1
)
, find
min
{
a
2
,
a
3
,
…
,
a
p
−
1
}
\min\{a_2, a_3, \dots, a_{p-1}\}
min
{
a
2
,
a
3
,
…
,
a
p
−
1
}
.
number theory