2014 IMO Problems
Contents
Problem 3
Points and
lie on side
of acute-angled
so that
and
. Points
and
lie on lines
and
, respectively, such that
is the midpoint of
, and
is the midpoint of
. Prove that lines
and
intersect on the circumcircle of
.
Problem 4
Points and
lie on side
of acute-angled
so that
and
. Points
and
lie on lines
and
, respectively, such that
is the midpoint of
, and
is the midpoint of
. Prove that lines
and
intersect on the circumcircle of
.
Problem 5
For each positive integer , the Bank of Cape Town issues coins of denomination
. Given a finite collection of such coins (of not necessarily different denominations) with total value at most
, prove that it is possible to split this collection into
or fewer groups, such that each group has total value at most
.
Problem 6
A set of lines in the plane is in 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 are; we call these its
. Prove that for all sufficiently large
, in any set of
lines in general position it is possible to colour at least
of the lines blue in such a way that none of its finite regions has a completely blue boundary.