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IMO Longlists
1974 IMO Longlists
36
All binomials are even [ILL 1974]
All binomials are even [ILL 1974]
Source:
January 3, 2011
ARML
binomial coefficients
Problem Statement
Consider the binomial coefficients
(
n
k
)
=
n
!
k
!
(
n
−
k
)
!
(
k
=
1
,
2
,
…
n
−
1
)
\binom{n}{k}=\frac{n!}{k!(n-k)!}\ (k=1,2,\ldots n-1)
(
k
n
)
=
k
!
(
n
−
k
)!
n
!
(
k
=
1
,
2
,
…
n
−
1
)
. Determine all positive integers
n
n
n
for which
(
n
1
)
,
(
n
2
)
,
…
,
(
n
n
−
1
)
\binom{n}{1},\binom{n}{2},\ldots ,\binom{n}{n-1}
(
1
n
)
,
(
2
n
)
,
…
,
(
n
−
1
n
)
are all even numbers.
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