Found problems: 181
Let $ k,m,n$ be integers such that $ 1 < n \leq m \minus{} 1 \leq k.$ Determine the maximum size of a subset $ S$ of the set $ \{1,2,3, \ldots, k\minus{}1,k\}$ such that no $ n$ distinct elements of $ S$ add up to $ m.$
155 birds $ P_1, \ldots, P_{155}$ are sitting down on the boundary of a circle $ C.$ Two birds $ P_i, P_j$ are mutually visible if the angle at centre $ m(\cdot)$ of their positions $ m(P_iP_j) \leq 10^{\circ}.$ Find the smallest number of mutually visible pairs of birds, i.e. minimal set of pairs $ \{x,y\}$ of mutually visible pairs of birds with $ x,y \in \{P_1, \ldots, P_{155}\}.$ One assumes that a position (point) on $ C$ can be occupied simultaneously by several birds, e.g. all possible birds.
Let $ S \equal{} \{1,2,3,\cdots ,280\}$. Find the smallest integer $ n$ such that each $ n$-element subset of $ S$ contains five numbers which are pairwise relatively prime.
A country has $100$ cities and $n$ airplane companies which take care of a total of $2018$ two-way direct flights between pairs of cities. There is a pair of cities such that one cannot reach one from the other with just one or two flights. What is the largest possible value of $n$ for which between any two cities there is a route (a sequence of flights) using only one of the airplane companies?
Twenty-one girls and twenty-one boys took part in a mathematical competition. It turned out that each contestant solved at most six problems, and for each pair of a girl and a boy, there was at least one problem that was solved by both the girl and the boy. Show that there is a problem that was solved by at least three girls and at least three boys.
The colonizers of a spherical planet have decided to build $N$ towns, each having area $1/1000$ of the total area of the planet. They also decided that any two points belonging to different towns will have different latitude and different longitude. What is the maximal value of $N$?