2016 EGMO Problems
Contents
[hide]Day 1
Problem 1
Let be an odd positive integer, and let be non-negative real numbers. Show thatwhere .
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.