## IMO 2014 Problem 6

A set of lines in the plane is in general position if no two are parallel and no three pass through the same point. A set of lines in general position cuts the plane into regions, some of which have finite area; we call these its finite regions. Prove that for all sufficiently large ${n}$, in any set of ${n}$ lines in general position it is possible to colour at least ${\sqrt{n}}$ lines blue in such a way that none of its finite regions has a completely blue boundary.

Note: Results with ${\sqrt{n}}$ replaced by ${c\sqrt{n}}$ will be awarded points depending on the value of the constant ${c}$.

IMO 2014 Problem 6 (Day 2)

## IMO 2014 Problem 5

For every positive integer ${n}$, Cape Town Bank issues some coins that has ${\frac{1}{n}}$ value. Let a collection of such finite coins (coins does not neccesarily have different values) which sum of their value is less than ${99+\frac{1}{2}}$. Prove that we can divide the collection into at most 100 groups such that sum of all coins’ value does not exceed 1.

IMO 2014 Problem 5 (Day 2)

Categories: Combinatorics, IMO, Olympiad Tags: ,

## IMO 2014 Problem 4

Let ${P}$ and ${Q}$ be on segment ${BC}$ of an acute triangle ${ABC}$ such that ${\angle PAB=\angle BCA}$ and ${\angle CAQ=\angle ABC}$. Let ${M}$ and ${N}$ be the points on ${AP}$ and ${AQ}$, respectively, such that ${P}$ is the midpoint of ${AM}$ and ${Q}$ is the midpoint of ${AN}$. Prove that the intersection of ${BM}$ and ${CN}$ is on the circumference of triangle ${ABC}$.

IMO 2014 Problem 4 (Day 2)

Categories: Geometry, IMO, Olympiad Tags: ,

## IMO 2014 Problem 3

Convex quadrilateral ${ABCD}$ has ${\angle ABC = \angle CDA = 90^\circ}$. Point ${H}$ is the foot of the perpendicular from ${A}$ to ${BD}$. Points ${S}$ and ${T}$ lie on sides ${AB}$ and ${AD}$, respectively, such that ${H}$ lies inside triangle ${SCT}$ and

$\displaystyle \angle CHS -\angle CSB = 90^\circ,\ \angle THC-\angle DTC = 90^\circ.$

Prove that the line ${BD}$ is tangent to the circumcircle of triangle ${TSH}$.

IMO 2014 Problem 3 (Day 1)

Categories: Geometry, IMO, Problem Solving Tags: ,

## IMO 2014 Problem 2

Let ${n \geq 2}$ be an integer. Consider a ${n \times n}$ chessboard consisting of ${n^2}$ unit squares. A configuration of ${n}$ rooks on this board is peaceful if every row and every column contains exactly one rook. Find the greatest positive integer ${k}$ such that for each peaceful configuration of ${n}$ rooks, there is a ${k \times k}$ square which does not contain a rook on any of its ${k^2}$ unit squares.

IMO 2014 Problem 2 (Day 1)

## IMO 2014 Problem 1

Let ${a_0 be an infinite sequence of positive integers. Prove that there exists a unique integer ${n \geq 1}$ such that

$\displaystyle a_n<\frac{a_0+a_1+...+a_n}{n} \leq a_{n+1}.$

IMO 2014 problem 1 (Day 1)

Let ${G}$ be a finite group such that ${|G|=mp^a}$ where ${p}$ is a prime number, ${a \geq 1}$ and ${\gcd(m,p)=1}$. Then there exists a subgroup ${H \leq G}$ such that ${|H|=p^a}$. (such a subgroup is called a Sylow subgroup).