2020 USAMO Problems
Let be a fixed acute triangle inscribed in a circle with center . A variable point is chosen on minor arc of , and segments and meet at . Denote by and the circumcenters of triangles and , respectively. Determine all points for which the area of triangle is minimized.
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 an odd prime. An integer is called a quadratic non-residue if does not divide for any integer .
Denote by the set of all integers such that , and both and are quadratic non-residues. Calculate the remainder when the product of the elements of is divided by .
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.
A finite set of points in the coordinate plane is called overdetermined if and there exists a nonzero polynomial , with real coefficients and of degree at most , satisfying for every point .
For each integer , find the largest integer (in terms of ) such that there exists a set of distinct points that is not overdetermined, but has overdetermined subsets.
Let be an integer. Let and be real numbers such that Prove that
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