2016 EGMO Problems
Contents
[hide]Day 1
Problem 1
Let be an odd positive integer, and let
be non-negative real numbers. Show that
where
.
Problem 2
Let be a cyclic quadrilateral, and let diagonals
and
intersect at
.Let
and
be the midpoints of segments
and
, respecctively. Lines
and
intersect at
, and line
intersects diagonals
and
at different points
and
, respectively. Prove that line
is tangent to the circle through
and
.
Problem 3
Let be a positive integer. Consider a
array of square unit cells. Two different cells are related to each other if they are in either the same row or in the same column.No cell is related to itself.Some cells are coloured blue, such that every cell is related to at lest two blue cells.Determine the minimum number of blue cells.
Day 2
Problem 4
Two circles and
, of equal radius intersect at different points
and
. Consider a circle
externally tangent to
at
and internally tangent to
at point
. Prove that lines
and
intersect at a point lying on
.
Problem 5
Let and
be integers such that
and
. Place rectangular tiles, each of size
, or
on a
chessboard so that each tile covers exactly
cells and no two tiles overlap. Do this until no further tile can be placed in this way. For each such
and
, determine the minimum number of tiles that such an arrangement may contain.
Problem 6
Let be the set of all positive integers
such that
has a divisor in the range
. Prove that there are infinitely many elements of
of each of the forms
and no elements of
of the form
and
, where
is an integer.