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| − | ==Day 1==
| + | #redirect [[2020 USAMO Problems]] |
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| − | ===Problem 1===
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| − | Let <math>ABC</math> be a fixed acute triangle inscribed in a circle <math>\omega</math> with center <math>O</math>. A variable point <math>X</math> is chosen on minor arc <math>AB</math> of <math>\omega</math>, and segments <math>CX</math> and <math>AB</math> meet at <math>D</math>. Denote by <math>O_1</math> and <math>O_2</math> the circumcenters of triangles <math>ADX</math> and <math>BDX</math>, respectively. Determine all points <math>X</math> for which the area of triangle <math>OO_1O_2</math> is minimized.
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| − | [[2020 USOMO Problems/Problem 1|Solution]] | |
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| − | ===Problem 2===
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| − | An empty <math>2020 \times 2020 \times 2020</math> cube is given, and a <math>2020 \times 2020</math> grid of square unit cells is drawn on each of its six faces. A beam is a <math>1 \times 1 \times 2020</math> rectangular prism. Several beams are placed inside the cube subject to the following conditions:
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| − | <math>\bullet</math> The two <math>1 \times 1</math> faces of each beam coincide with unit cells lying on opposite faces
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| − | of the cube. (Hence, there are <math>3 \cdot 2020^2</math> possible positions for a beam.)
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| − | <math>\bullet</math> No two beams have intersecting interiors.
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| − | <math>\bullet</math> The interiors of each of the four <math>1 \times 2020</math> faces of each beam touch either a face of the cube or the interior of the face of another beam.
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| − | What is the smallest positive number of beams that can be placed to satisfy these conditions?
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| − | [[2020 USOMO Problems/Problem 2|Solution]]
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| − | ===Problem 3===
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| − | Let <math>p</math> be an odd prime. An integer <math>x</math> is called a <i>quadratic non-residue</i> if
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| − | <math>p</math> does not divide <math>x - t^2</math> for any integer <math>t</math>.
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| − | Denote by <math>A</math> the set of all integers <math>a</math> such that <math>1 \le a < p</math>, and both <math>a</math> and <math>4 - a</math> are
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| − | quadratic non-residues. Calculate the remainder when the product of the elements of <math>A</math>
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| − | is divided by <math>p</math>.
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| − | [[2020 USOMO Problems/Problem 3|Solution]]
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| − | ==Day 2==
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