AoPS Academy
such that 
and 
are coprime if and only if 
?
be a positive integer. Suppose given any real 
with decimal representation 
, one can color the digits 
with colors so that the following hold:
except those of this color, the resulting decimal number is rational.. be a positive integer. Consider a 
array of square unit cells. Two corner cells that lie on the same longest diagonal are colored black, and the rest of the array is white. A move consists of choosing a row or a column and changing the color of every cell in the chosen row or column.
the equilateral triangles 
, 
and 
are constructed. Given the acute triangle 
, construct the triangle .
the equilateral triangles , and are constructed. Given the acute triangle , construct the triangle .
satisfying
Prove
