MathDB
Problems
Contests
International Contests
Kvant Problems
Kvant 2020
M2624
M2624
Part of
Kvant 2020
Problems
(1)
Inequality on integers [Poland 2017, P3]
Source: Polish Mathematical Olympiad Finals, Problem 3
4/4/2017
Integers
a
1
,
a
2
,
…
,
a
n
a_1, a_2, \ldots, a_n
a
1
,
a
2
,
…
,
a
n
satisfy
1
<
a
1
<
a
2
<
…
<
a
n
<
2
a
1
.
1<a_1<a_2<\ldots < a_n < 2a_1.
1
<
a
1
<
a
2
<
…
<
a
n
<
2
a
1
.
If
m
m
m
is the number of distinct prime factors of
a
1
a
2
⋯
a
n
a_1a_2\cdots a_n
a
1
a
2
⋯
a
n
, then prove that
(
a
1
a
2
⋯
a
n
)
m
−
1
≥
(
n
!
)
m
.
(a_1a_2\cdots a_n)^{m-1}\geq (n!)^m.
(
a
1
a
2
⋯
a
n
)
m
−
1
≥
(
n
!
)
m
.
number theory
Poland