IMO 2014 Problem 6

July 9, 2014 Leave a 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 Leave a 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 2014 Problem 2

July 8, 2014 Leave a comment

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

July 8, 2014 Leave a comment

Let {a_0<a_1<...} 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)

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Categories: Algebra, IMO, Olympiad Tags:

Existence of Sylow subgroups

June 19, 2014 Leave a comment

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).

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