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Contests
National and Regional Contests
China Contests
China Team Selection Test
2024 China Team Selection Test
7
7
Part of
2024 China Team Selection Test
Problems
(1)
sum of reciprocals
Source: 2024 CTST P7
3/11/2024
For coprime positive integers
a
,
b
a,b
a
,
b
,denote
(
a
−
1
m
o
d
b
)
(a^{-1}\bmod{b})
(
a
−
1
mod
b
)
by the only integer
0
≤
m
<
b
0\leq m<b
0
≤
m
<
b
such that
a
m
≡
1
(
m
o
d
b
)
am\equiv 1\pmod{b}
am
≡
1
(
mod
b
)
(1)Prove that for pairwise coprime integers
a
,
b
,
c
a,b,c
a
,
b
,
c
,
1
<
a
<
b
<
c
1<a<b<c
1
<
a
<
b
<
c
,we have
(
a
−
1
m
o
d
b
)
+
(
b
−
1
m
o
d
c
)
+
(
c
−
1
m
o
d
a
)
>
a
.
(a^{-1}\bmod{b})+(b^{-1}\bmod{c})+(c^{-1}\bmod{a})>\sqrt a.
(
a
−
1
mod
b
)
+
(
b
−
1
mod
c
)
+
(
c
−
1
mod
a
)
>
a
.
(2)Prove that for any positive integer
M
M
M
,there exists pairwise coprime integers
a
,
b
,
c
a,b,c
a
,
b
,
c
,
M
<
a
<
b
<
c
M<a<b<c
M
<
a
<
b
<
c
such that
(
a
−
1
m
o
d
b
)
+
(
b
−
1
m
o
d
c
)
+
(
c
−
1
m
o
d
a
)
<
100
a
.
(a^{-1}\bmod{b})+(b^{-1}\bmod{c})+(c^{-1}\bmod{a})< 100\sqrt a.
(
a
−
1
mod
b
)
+
(
b
−
1
mod
c
)
+
(
c
−
1
mod
a
)
<
100
a
.
number theory