Difference between revisions of "2016 EGMO Problems"
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Latest revision as of 13:49, 24 December 2022
Contents
Day 1
Problem 1
Let 
be an odd positive integer, and let 
be non-negative real numbers. Show that![\[\min_{i=1,\ldots,n} (x_i^2+x_{i+1}^2) \leq \max_{j=1,\ldots,n} (2x_jx_{j+1})\]](http://latex.artofproblemsolving.com/f/b/2/fb2f47fccd0301bc7219c5c7d8238088d8fec025.png)
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.
