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Hungary Contests
Eotvos Mathematical Competition (Hungary)
1906 Eotvos Mathematical Competition
3
3
Part of
1906 Eotvos Mathematical Competition
Problems
(1)
product (a_1 - 1)(a_2 - 2) ... (a_n -n) is an even number
Source: Eotvos 1906 p3
9/6/2024
Let
a
1
,
a
2
,
.
.
.
,
a
n
a_1, a_2, ...,a_n
a
1
,
a
2
,
...
,
a
n
represent an arbitrary arrangement of the numbers
1
,
2
,
.
.
.
,
n
1, 2, ...,n
1
,
2
,
...
,
n
. Prove that, if
n
n
n
is odd, the product
(
a
1
−
1
)
(
a
2
−
2
)
.
.
.
(
a
n
−
n
)
(a_1 - 1)(a_2 - 2) ... (a_n -n)
(
a
1
−
1
)
(
a
2
−
2
)
...
(
a
n
−
n
)
is an even number.
number theory
Even
permutation