2020 USOMO Problems
Day 1
Problem 1
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.
Problem 2
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:
