2017 IMO Problems
For each integer , define the sequence for as Determine all values of such that there exists a number such that for infinitely many values of .
Let be the set of real numbers , determine all functions such that for any real numbers and =
A hunter and an invisible rabbit play a game in the Euclidean plane. The rabbit's starting point, , and the hunter's starting point, , are the same. After rounds of the game, the rabbit is at point and the hunter is at point . In the nth round of the game, three things occur in order.
(i) The rabbit moves invisibly to a point such that the distance between and is exactly 1.
(ii) A tracking device reports a point to the hunter. The only guarantee provided by the tracking device is that the distance between and is at most 1.
(iii) The hunter moves visibly to a point such that the distance between and is exactly 1.
Is it always possible, no matter how the rabbit moves, and no matter what points are reported by the tracking device, for the hunter to choose her moves so that after rounds she can ensure that the distance between her and the rabbit is at most 100?
Let and be different points on a circle such that is not a diameter. Let be the tangent line to at . Point is such that is the midpoint of the line segment . Point is chosen on the shorter arc of so that the circumcircle of triangle intersects at two distinct points. Let be the common point of and that is closer to . Line meets again at . Prove that the line is tangent to .
An integer is given. A collection of soccer players, no two of whom are of the same height, stand in a row. Sir Alex wants to remove players from this row leaving a new row of players in which the following conditions hold:
() no one stands between the two tallest players,
() no one stands between the third and fourth tallest players,
() no one stands between the two shortest players.
Show that this is always possible.
An ordered pair of integers is a primitive point if the greatest common divisor of and is . Given a finite set of primitive points, prove that there exist a positive integer and integers such that, for each in , we have:
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