2020 USOJMO Problems

Day 1

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.

Problem 1

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.

Solution

Problem 2

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 .

Solution

Problem 3

An empty cube is given, and a grid of square unit cells is drawn on each of its six faces. A [i]beam[/i] 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?

Solution