Having won the elections in Linetown, Turbo the Snail has become mayor, and one of the most pressing issues he needs to work on is the road network. Linetown can be represented as a configuration of lines
in the plane, of which no two are parallel and no three are concurrent.

There is one house in Linetown for each pairwise intersection of two lines. The lines are used as roads by the townsfolk. In the past, the roads in Linetown used to be two-way, but this often led to residents accidentally cycling back to where they started.

Turbo wants to make each of the roads one-way such that it is impossible for any resident to start at a house, follow the roads in the correct directions, and end up back at the original house. In how many ways can Turbo achieve this?

Proposed by Archit Manas

Let be real numbers satisfying Prove that and determine all cases of equality.

Proposed by Shantanu Nene

Let and Prove that
Let and Prove that
Let and Prove that

The Bank of Oslo issues two types of coin: aluminum (denoted A) and bronze (denoted B). Marianne has aluminum coins and bronze coins arranged in a row in some arbitrary initial order. A chain is any subsequence of consecutive coins of the same type. Given a fixed positive integer , Gilberty repeatedly performs the following operation: he identifies the longest chain containing the coin from the left and moves all coins in that chain to the left end of the row. For example, if and , the process starting from the ordering would be

Find all pairs with such that for every initial ordering, at some moment during the process, the leftmost coins will all be of the same type.

Let be a fixed integer. There are beads on a circular necklace. You wish to paint the beads using colors, such that among any consecutive beads every color appears at least once. Find the largest value of for which this task is possible.

Carl Schildkraut, USA

In triangle , let , , and denote the feet of the altitudes from , , and , respectively, and let denote the circumcenter of . Points and denote the projections of and , respectively, onto , and . There exists a point such that and . If and lies on such that , prove that line bisects .