CJMO 2021

Let $C_1$ and $C_2$ be two concentric circles with $C_1$ inside $C_2$ . Let $P_1$ and $P_2$ be two points on $C_1$ that are not diametrically opposite. Extend the segment $P_1P_2$ past $P_2$ until it meets the circle $C_2$ in $Q_2$ . The tangent to $C_2$ at $Q_2$ and the tangent to $C_1$ at $P_1$ meet in a point $X$ . Draw from X the second tangent to $C_2$ which meets $C_2$ at the point $Q_1$ . Show that $P_1X$ bisects angle $Q_1P_1Q_2$ .

How many ways are there to permute the first $n$ positive integers such that in the permutation, for each value of $k \le n$ , the first $k$ elements of the permutation have distinct remainder mod $k$ ?