2021 IMO Problems
Let be an integer. Ivan writes the numbers each on different cards. He then shuffles these cards, and divides them into two piles. Prove that at least one of the piles contains two cards such that the sum of their numbers is a perfect square.
Show that the inequality holds for all real numbers .
Let be an interior point of the acute triangle with so that . The point on the segment satisfies , the point on the segment satisfies , and the point on the line satisfies . Let and be the circumcenters of the triangles and respectively. Prove that the lines , , and are concurrent.
Let be a circle with centre , and a convex quadrilateral such that each of the segments and is tangent to . Let be the circumcircle of the triangle . The extension of beyond meets at , and the extension of beyond meets at . The extensions of and beyond meet at and , respectively. Prove that
Two squirrels, Bushy and Jumpy, have collected 2021 walnuts for the winter. Jumpy numbers the walnuts from 1 through 2021, and digs 2021 little holes in a circular pattern in the ground around their favourite tree. The next morning Jumpy notices that Bushy had placed one walnut into each hole, but had paid no attention to the numbering. Unhappy, Jumpy decides to reorder the walnuts by performing a sequence of 2021 moves. In the -th move, Jumpy swaps the positions of the two walnuts adjacent to walnut .
Prove that there exists a value of such that, on the -th move, Jumpy swaps some walnuts and such that .
Let be an integer, be a finite set of (not necessarily positive) integers, and be subsets of . Assume that for each the sum of the elements of is . Prove that contains at least elements.
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