MathDB
Problems
Contests
National and Regional Contests
China Contests
China Team Selection Test
2024 China Team Selection Test
12
12
Part of
2024 China Team Selection Test
Problems
(1)
Difficult NT
Source: 2024 CTST P12
3/12/2024
Given positive odd number
m
m
m
and integer
a
.
{a}.
a
.
Proof: For any real number
c
,
c,
c
,
#
{
x
∈
Z
∩
[
c
,
c
+
m
]
∣
x
2
≡
a
(
m
o
d
m
)
}
≤
2
+
log
2
m
.
\#\left\{x\in\mathbb Z\cap [c,c+\sqrt m]\mid x^2\equiv a\pmod m\right\}\le 2+\log_2m.
#
{
x
∈
Z
∩
[
c
,
c
+
m
]
∣
x
2
≡
a
(
mod
m
)
}
≤
2
+
lo
g
2
m
.
Proposed by Yinghua Ai
number theory
2024 CTST