Olympiad problems

2018 Serbia National Math Olympiad, 3

Let $n$ be a positive integer. There are given $n$ lines such that no two are parallel and no three meet at a single point. a) Prove that there exists a line such that the number of intersection points of these $n$ lines on both of its sides is at least $$\left \lfloor \frac{(n-1)(n-2)}{10} \right \rfloor.$$ Notice that the points on the line are not counted. b) Find all $n$ for which there exists a configurations where the equality is achieved.

2007 Alexandru Myller, 2

Tags: inequalities , discrete geometry , plane geometry , angle , geometry

$ n $ lines meet at a point. Each one of the $ 2n $ disjoint angles formed around this point by these lines has either $ 7^{\circ} $ or $ 17^{\circ} . $ [b]a)[/b] Find $ n. $ [b]b)[/b] Prove that among these lines there are at least two perpendicular ones.

2017 Stars of Mathematics, 4

Tags: discrete geometry , hungary

Let be distinct points on a plane, four of which form a quadrangle, and three of which are in the interior or boundary of this quadrangle. Show that the diagonals of this quadrangle are longer than the double of the minimum of the distances between any two of these seven points. [i]Paul Erdős[/i] [hide=Side note]If the quadrangle is convex, the constant from the inequality can be improved from $ 2 $ to $ \sqrt{\frac{3\pi}{2}}. $[/hide]

2019 Romania National Olympiad, 2

Tags: geometry , trapezoid , combinatorics , complex number geometry , discrete geometry

Find the number of trapeziums that it can be formed with the vertices of a regular polygon.

2006 Mathematics for Its Sake, 3

Tags: geometry , discrete math , discrete geometry , g.m. , pigeonhole principle

Show that if the point $ M $ is situated in the interior of a square $ ABCD, $ then, among the segments $ MA,MB,MC,MD, $ [b]a)[/b] at most one of them is greater with a factor of $ \sqrt 5/2 $ than the side of the square. [b]b)[/b] at most two of them are greater than the side of the square. [b]c)[/b] at most three of them are greater with a factor of $ \sqrt 2/2 $ than the side of the square.

2015 IFYM, Sozopol, 8

Tags: set theory , discrete geometry

The points $A_1,A_...,A_n$ lie on a circle with radius 1. The points $B_1,B_2,…,B_n$ are such that $B_i B_j<A_i A_j$ for $i\neq j$. Is it always true that the points $B_1,B_2,...,B_n$ lie on a circle with radius lesser than 1?

2017 Canada National Olympiad, 5

Tags: combinatorial geometry , discrete geometry

There are $100$ circles of radius one in the plane. A triangle formed by the centres of any three given circles has area at most $2017$. Prove that there is a line intersecting at least three of the circles.