Olympiad problems

2015 IMAR Test, 2

Let $n$ be a positive integer and let $G_n$ be the set of all simple graphs on $n$ vertices. For each vertex $v$ of a graph in $G_n$, let $k(v)$ be the maximal cardinality of an independent set of neighbours of $v$. Determine $max_{G \in G_n} \Sigma_{v\in V (G)}k(v)$ and the graphs in $G_n$ that achieve this value.

2013 Bosnia and Herzegovina Junior BMO TST, 4

Tags: combinatorics , polygon , vertices

It is given polygon with $2013$ sides $A_{1}A_{2}...A_{2013}$. His vertices are marked with numbers such that sum of numbers marked by any $9$ consecutive vertices is constant and its value is $300$. If we know that $A_{13}$ is marked with $13$ and $A_{20}$ is marked with $20$, determine with which number is marked $A_{2013}$

2010 Sharygin Geometry Olympiad, 2

Tags: circumcircle , geometry , vertices

Two intersecting triangles are given. Prove that at least one of their vertices lies inside the circumcircle of the other triangle. (Here, the triangle is considered the part of the plane bounded by a closed three-part broken line, a point lying on a circle is considered to be lying inside it.)

1978 Bundeswettbewerb Mathematik, 2

Tags: geometry , square , area , vertices , triangle

Seven distinct points are given inside a square with side length $1.$ Together with the square's vertices, they form a set of $11$ points. Consider all triangles with vertices in $M.$ a) Show that at least one of these triangles has an area not exceeding $1\slash 16.$ b) Give an example in which no four of the seven points are on a line and none of the considered triangles has an area of less than $1\slash 16.$