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IMO 2016 – Problem 1

July 26, 2016 Leave a comment

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.

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IMO 2015 Problem 1

July 10, 2015 Leave a comment

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

Categories: Olympiad Tags: , ,

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

July 9, 2014 Leave a comment

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

July 8, 2014 Leave a comment

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

December 18, 2013 Leave a comment

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}

is minimal.

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}.

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