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China Team Selection Test
2024 China Team Selection Test
12
Difficult NT
Difficult NT
Source: 2024 CTST P12
March 12, 2024
number theory
2024 CTST
Problem Statement
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
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