2016 IMO Problems
Triangle has a right angle at . Let be the point on line such that and lies between and . Point is chosen so that and is the bisector of . Point is chosen so that and is the bisector of . Let be the midpoint of . Let be the point such that is a parallelogram. Prove that and are concurrent.
Find all integers for which each cell of table can be filled with one of the letters and in such a way that:
Note. The rows and columns of an table are each labelled to in a natural order. Thus each cell corresponds to a pair of positive integer with . For , the table has diagonals of two types. A diagonal of first type consists all cells for which is a constant, and the diagonal of this second type consists all cells for which is constant.
Let be a convex polygon in the plane. The vertices have integral coordinates and lie on a circle. Let be the area of . An odd positive integer is given such that the squares of the side lengths of are integers divisible by . Prove that is an integer divisible by .
A set of positive integers is called fragrant if it contains at least two elements and each of its elements has a prime factor in common with at least one of the other elements. Let . What is the least possible positive integer value of such that there exists a non-negative integer for which the set is fragrant?
is written on the board, with linear factors on each side. What is the least possible value of for which it is possible to erase exactly of these linear factors so that at least one factor remains on each side and the resulting equation has no real solutions?
There are line segments in the plane such that every two segments cross and no three segments meet at a point. Geoff has to choose an endpoint of each segment and place a frog on it facing the other endpoint. Then he will clap his hands times. Every time he claps,each frog will immediately jump forward to the next intersection point on its segment. Frogs never change the direction of their jumps. Geoff wishes to place the frogs in such a way that no two of them will ever occupy the same intersection point at the same time.
(a) Prove that Geoff can always fulfill his wish if is odd.
(b) Prove that Geoff can never fulfill his wish if is even.
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