MathDB
Problems
Contests
National and Regional Contests
China Contests
China Team Selection Test
2023 China Team Selection Test
P3
P3
Part of
2023 China Team Selection Test
Problems
(1)
Very interesting NT
Source: China TST 2023, Test 1, Problem 3
3/14/2023
(1) Let
a
,
b
a,b
a
,
b
be coprime positive integers. Prove that there exists constants
λ
\lambda
λ
and
β
\beta
β
such that for all integers
m
m
m
,
∣
∑
k
=
1
m
−
1
{
a
k
m
}
{
b
k
m
}
−
λ
m
∣
≤
β
\left| \sum\limits_{k=1}^{m-1} \left\{ \frac{ak}{m} \right\}\left\{ \frac{bk}{m} \right\} - \lambda m \right| \le \beta
k
=
1
∑
m
−
1
{
m
ak
}
{
m
bk
}
−
λm
≤
β
(2) Prove that there exists
N
N
N
such that for all
p
>
N
p>N
p
>
N
(where
p
p
p
is a prime number), and any positive integers
a
,
b
,
c
a,b,c
a
,
b
,
c
positive integers satisfying
p
∤
(
a
+
b
)
(
b
+
c
)
(
c
+
a
)
p\nmid (a+b)(b+c)(c+a)
p
∤
(
a
+
b
)
(
b
+
c
)
(
c
+
a
)
, there are at least
⌊
p
12
⌋
\lfloor \frac{p}{12} \rfloor
⌊
12
p
⌋
solutions
k
∈
{
1
,
⋯
,
p
−
1
}
k\in \{1,\cdots,p-1\}
k
∈
{
1
,
⋯
,
p
−
1
}
such that
{
a
k
p
}
+
{
b
k
p
}
+
{
c
k
p
}
≤
1
\left\{\frac{ak}{p}\right\} + \left\{\frac{bk}{p}\right\} + \left\{\frac{ck}{p}\right\} \le 1
{
p
ak
}
+
{
p
bk
}
+
{
p
c
k
}
≤
1
China TST
number theory