1976 AMC 12 Question 23 - thichhoctoan.com

Question 23

For integers $k$ and $n$ such that $1\le k\lt n,$ let $$
\binom nk=\frac{n!}{k!(n-k)!}.
$$ Then $\left(\frac{n-2k-1}{k+1}\right)\binom nk$ is an integer

for all $k$ and $n$
for all even values of $k$ and $n,$ but not for all $k$
and $n$
for all odd values of $k$ and $n,$ but not for all $k$
and $n$
if $k=1$ or $n-1,$ but not for all odd values of $k$ and
$n$
if $n$ is divisible by $k,$ but not for all even values of
$k$ and $n$