Question 47
Let $k$ be an integer, $1 \lt k \lt 100$. For every permutation $a_{1}, a_{2}, \ldots, a_{100}$ of the integers $1, 2, \ldots, 100$, set $a_{101} = 0$ and choose the least integer $m \gt k$ such that $a_{m}$ is less than at least $k-1$ of the numbers $a_{1}, a_{2}, \ldots, a_{k}$. Find all $k$ for which the number of permutations with $a_{m} = 1$, is equal to $\frac{100!}{4}$.
Peter Boyvalenkov, Emil Kolev, Nikolai Nikolov
Peter Boyvalenkov, Emil Kolev, Nikolai Nikolov