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Problems
Contests
National and Regional Contests
China Contests
China Team Selection Test
2024 China Team Selection Test
16
16
Part of
2024 China Team Selection Test
Problems
(1)
NT with condition that is expected to be almost always true by PNT
Source: 2024 CTST P16
3/25/2024
m
>
1
m>1
m
>
1
is an integer such that
[
2
m
−
m
+
1
,
2
m
]
[2m-\sqrt{m}+1, 2m]
[
2
m
−
m
+
1
,
2
m
]
contains a prime. Prove that for any pairwise distinct positive integers
a
1
a_1
a
1
,
a
2
a_2
a
2
,
…
\dots
…
,
a
m
a_m
a
m
, there is always
1
≤
i
,
j
≤
m
1\leq i,j\leq m
1
≤
i
,
j
≤
m
such that
a
i
(
a
i
,
a
j
)
≥
m
\frac{a_i}{(a_i, a_j)}\geq m
(
a
i
,
a
j
)
a
i
≥
m
.
number theory