2021年USAJMO 真题
Day 1
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 denote the set of positive integers. Find all functions
such that for positive integers
and
Problem 2
Rectangles
and
are erected outside an acute triangle
Suppose that
Prove that lines
and
are concurrent.
Problem 3
An equilateral triangle of side length
is given. Suppose that
equilateral triangles with side length 1 and with non-overlapping interiors are drawn inside
, such that each unit equilateral triangle has sides parallel to
, but with opposite orientation. (An example with
is drawn below.)
Prove that
Day 2
Problem 4
Carina has three pins, labeled , and
, respectively, located at the origin of the coordinate plane. In a move, Carina may move a pin to an adjacent lattice point at distance
away. What is the least number of moves that Carina can make in order for triangle
to have area 2021?
(A lattice point is a point in the coordinate plane where
and
are both integers, not necessarily positive.)
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