2020 USOJMO Problems
Note: For any geometry problem whose statement begins with an asterisk , the first page of the solution must be a large, in-scale, clearly labeled diagram. Failure to meet this requirement will result in an automatic 1-point deduction.
Let be an integer. Carl has books arranged on a bookshelf. Each book has a height and a width. No two books have the same height, and no two books have the same width. Initially, the books are arranged in increasing order of height from left to right. In a move, Carl picks any two adjacent books where the left book is wider and shorter than the right book, and swaps their locations. Carl does this repeatedly until no further moves are possible. Prove that regardless of how Carl makes his moves, he must stop after a finite number of moves, and when he does stop, the books are sorted in increasing order of width from left to right.
Let be the incircle of a fixed equilateral triangle . Let be a variable line that is tangent to and meets the interior of segments and at points and , respectively. A point is chosen such that and . Find all possible locations of the point , over all choices of .
An empty cube is given, and a grid of square unit cells is drawn on each of its six faces. A beam is a rectangular prism. Several beams are placed inside the cube subject to the following conditions:
- The two faces of each beam coincide with unit cells lying on opposite faces of the cube. (Hence, there are possible positions for a beam.)
- No two beams have intersecting interiors.
- The interiors of each of the four faces of each beam touch either a face of the cube or the interior of the face of another beam.
What is the smallest positive number of beams that can be placed to satisfy these conditions?
Let be a convex quadrilateral inscribed in a circle and satisfying . Points and are chosen on sides and such that and . Prove that .
Suppose that are distinct ordered pairs of nonnegative integers. Let denote the number of pairs of integers satisfying and . Determine the largest possible value of over all possible choices of the ordered pairs.
Let be an integer. Let be a nonconstant -variable polynomial with real coefficients. Assume that whenever are real numbers, at least two of which are equal, we have . Prove that cannot be written as the sum of fewer than monomials. (A monomial is a polynomial of the form , where is a nonzero real number and , , , are nonnegative integers.)
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