2018 IMO Problems
Let be the circumcircle of acute triangle . Points and are on segments and respectively such that . The perpendicular bisectors of and intersect minor arcs and of at points and respectively. Prove that lines and are either parallel or they are the same line.
Find all numbers for which there exists real numbers satisfying and for
An anti-Pascal triangle is an equilateral triangular array of numbers such that, except for the numbers in the bottom row, each number is the absolute value of the difference of the two numbers immediately below it. For example, the following is an anti-Pascal triangle with four rows which contains every integer from to
Does there exist an anti-Pascal triangle with rows which contains every integer from to ?
A site is any point in the plane such that and are both positive integers less than or equal to 20. Initially, each of the 400 sites is unoccupied. Amy and Ben take turns placing stones with Amy going first. On her turn, Amy places a new red stone on an unoccupied site such that the distance between any two sites occupied by red stones is not equal to . On his turn, Ben places a new blue stone on any unoccupied site. (A site occupied by a blue stone is allowed to be at any distance from any other occupied site.) They stop as soon as a player cannot place a stone. Find the greatest such that Amy can ensure that she places at least red stones, no matter how Ben places his blue stones.
Let be an infinite sequence of positive integers. Suppose that there is an integer such that, for each , the number is an integer. Prove that there is a positive integer such that for all
A convex quadrilateral satisfies Point lies inside so that and Prove that .
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