Mathematics

$n$ people numbered $1$ to $n$ stand in a circle. Going around the circle starting from person 1, every second person is removed (first person 2, then 4, and so on, continuing around the circle) until one person remains. Find a formula for the position $J(n)$ of the survivor.

Arrange the numbers $1,1,2,2,\dots ,n,n$ in a row so that for each $k$ exactly $k$ other numbers stand between the two copies of $k$. For $n=3$ the sequence 2, 3, 1, 2, 1, 3 works. For which $n$ is such an arrangement possible?

100 prisoners numbered 1 to 100 face a room with 100 drawers, into which the numbers 1 to 100 have been placed in random order. One at a time, each prisoner enters, opens at most 50 drawers looking for their own number, and leaves the room exactly as it was, with no communication afterwards. All are…

A convex polyhedron has $V$ vertices, $E$ edges and $F$ faces. Prove that $V-E+F=2$.

A simple polygon has all of its vertices at points with integer coordinates. It has $i$ integer points in its interior and $b$ integer points on its boundary. Find a formula for its area in terms of $i$ and $b$, and prove it.

Assume birthdays are independent and uniformly distributed over 365 days. What is the smallest number of people for which the probability that at least two of them share a birthday exceeds 1/2?

Two players of equal skill play a series of fair rounds, and the first to win 6 rounds takes the whole stake. The game is interrupted when one player leads 5 rounds to 3. How should the stake be divided fairly?

A finite set of points in the plane does not lie on a single line. Prove that some line passes through exactly two of the points.

$n$ guests check their hats, and the hats are handed back in a uniformly random order. Let $p_n$ be the probability that no guest gets their own hat. Find a formula for $p_n$ and its limit as $n\to \infty$.

Fifteen schoolgirls go for a walk every day for a week, in five rows of three. Arrange the rows for the seven days so that no two girls walk in the same row more than once.