PL-OM 1995

How many subsets of $\{1, 2, ... , 2n\}$ do not contain two numbers with sum $2n+1$ ?

The diagonals of a convex pentagon divide it into a small pentagon and ten triangles. What is the largest number of the triangles that can have the same area?

The positive reals $x_1, x_2, ... , x_n$ have harmonic mean $1$ . Find the smallest possible value of $x_1 + \frac{x_2 ^2}{2} + \frac{x_3 ^3}{3} + ... + \frac{x_n ^n}{n}$ .

An urn contains $n$ balls labeled $1, 2, ... , n$ . We draw the balls out one by one (without replacing them) until we obtain a ball whose number is divisible by $k$ . Find all $k$ such that the expected number of balls removed is $k$ .