2021 USAMO Problems
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.
Rectangles and are erected outside an acute triangle Suppose thatProve that lines and are concurrent.
The Planar National Park is a subset of the Euclidean plane consisting of several trails which meet at junctions. Every trail has its two endpoints at two different junctions whereas each junction is the endpoint of exactly three trails. Trails only intersect at junctions (in particular, trails only meet at endpoints). Finally, no trails begin and end at the same two junctions.
A visitor walks through the park as follows: she begins at a junction and starts walking along a trail. At the end of that first trail, she enters a junction and turns left. On the next junction she turns right, and so on, alternating left and right turns at each junction. She does this until she gets back to the junction where she started. What is the largest possible number of times she could have entered any junction during her walk, over all possible layouts of the park?
Let be an integer. An board is initially empty. Each minute, you may perform one of three moves: If there is an L-shaped tromino region of three cells without stones on the board (see figure; rotations not allowed), you may place a stone in each of those cells. If all cells in a column have a stone, you may remove all stones from that column. If all cells in a row have a stone, you may remove all stones from that row. For which is it possible that, after some non-zero number of moves, the board has no stones?
A finite set of positive integers has the property that, for each and each positive integer divisor of , there exists a unique element satisfying . (The elements and could be equal.)
Given this information, find all possible values for the number of elements of .
Let be an integer. Find all positive real solutions to the following system of equations:
Let be a convex hexagon satisfying , , , and Let , , and be the midpoints of , , and . Prove that the circumcenter of , the circumcenter of , and the orthocenter of are collinear.
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