KURSCHAK 1968

In an infinite sequence of positive integers every element (starting with the second) is the harmonic mean of its neighbors. Show that all the numbers must be equal.

There are $4n$ segments of unit length inside a circle radius $n$ . Show that given any line $L$ there is a chord of the circle parallel or perpendicular to $L$ which intersects at least two of the $4n$ segments.

For each arrangement $X$ of $n$ white and $n$ black balls in a row, let $f(X)$ be the number of times the color changes as one moves from one end of the row to the other. For each $k$ such that $0 < k < n$ , show that the number of arrangements $X$ with $f(X) = n -k$ is the same as the number with $f(X) = n + k$ .