### Archive

Posts Tagged ‘IMO’

## IMO 2016 – Problem 1

IMO 2016, Problem 1. Triangle ${BCF}$ has a right angle at ${B}$. Let ${A}$ be the point on line ${CF}$ such that ${FA=FB}$ and ${F}$ lies between ${A}$ and ${C}$. Point ${D}$ is chosen such that ${DA=DC}$ and ${AC}$ is the bisector of ${\angle DAB}$. Point ${E}$ is chosen such that ${EA=ED}$ and ${AD}$ is the bisector of ${\angle EAC}$. Let ${M}$ be the midpoint of ${CF}$. Let ${X}$ be the point such that ${AMXE}$ is a parallelogram (where ${AM || EX}$ and ${AE || MX}$). Prove that the lines ${BD, FX}$ and ${ME}$ are concurrent.

## IMO 2015 Problem 1

Problem 1. We say that a finite set ${\mathcal{S}}$ of points in the plane is balanced if, for any two different points ${A}$ and ${B}$ in ${\mathcal{S}}$, there is a point ${C}$ in ${\mathcal{S}}$ such that ${AC=BC}$. We say that ${\mathcal{S}}$ is center-free if for any three different points ${A}$, ${B}$ and ${C}$ in ${\mathcal{S}}$, there is no points ${P}$ in ${\mathcal{S}}$ such that ${PA=PB=PC}$.

(a) Show that for all integers ${n\ge 3}$, there exists a balanced set having ${n}$ points.

(b) Determine all integers ${n\ge 3}$ for which there exists a balanced center-free set having ${n}$ points.

Problem 2. Find all triples of positive integers ${(a, b, c)}$ such that ${ab-c, bc-a, ca-b}$ are all powers of 2.

Problem 3. Let ${ABC}$ be an acute triangle with ${AB > AC}$. Let ${\Gamma }$ be its cirumcircle., ${H}$ its orthocenter, and ${F}$ the foot of the altitude from ${A}$. Let ${M}$ be the midpoint of ${BC}$. Let ${Q }$ be the point on ${ \Gamma }$ such that ${\angle HQA }$ and let ${K }$ be the point on ${\Gamma }$ such that ${\angle HKQ }$. Assume that the points ${A,B,C,K }$and ${Q }$ are all different and lie on ${\Gamma}$ in this order.

Prove that the circumcircles of triangles ${KQH }$ and ${FKM }$ are tangent to each other.

Source: AoPS

## IMO 2014 Problem 6

July 9, 2014 1 comment

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

July 9, 2014 1 comment

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 1981 Day 1

Problem 1. Let ${P}$ be a point inside a given triangle ${ABC}$ and denote ${D,E,F}$ the feet of the perpendiculars from ${P}$ to the lines ${BC,CA,AB}$ respectively. Find ${P}$ such that the quantity
$\displaystyle \frac{BC}{PD}+\frac{CA}{PE}+\frac{AB}{PF}$
Problem 2. Let ${1 \leq r\leq n}$ and consider all subsets of ${r}$ elements of the set ${\{1,2,..,n\}}$. Each of these subsets has a smallest member. Let ${F(n,r)}$ denote the arithmetic mean of these smallest numbers. Prove that
$\displaystyle F(n,r)=\frac{n+1}{r+1}.$
Problem 3. Determine the maximum value of ${m^3+n^3}$ where ${m,n \in \{1,2,..,1981\}}$ with ${(n^2-mn-m^2)^2=1}$.