AoPS Academy
cups labeled 
, where the 
-th cup has capacity liters. In total, there are liters of water distributed among these cups such that each cup contains an integer amount of water. In each step, we may transfer water from one cup to another. The process continues until either the source cup becomes empty or the destination cup becomes full.
Prove that from any configuration where each cup contains an integer amount of water, it is possible to reach a configuration in which each cup contains exactly 1 liter of water in at most 
steps.
Prove that in at most 
steps, one can go from any configuration with integer water amounts to any other configuration with the same property.
and 
for which there exists a natural number
such that the number 
is integer for all natural 
.
has 
as the midpoint of 
and orthocenter 
. Let 
be an arbitrary point on the nine point circle of 
. The line through perpendicular to 
intersects at 
. The line through 
perpendicular to 
intersects 
at 
. If 
is the midpoint of , show that 
.
, mark a point 
such that 
, where is the point of intersection of the rays 
and 
, and is the point of intersection of the rays 
and . Prove that the bisectors of the angles 
and 
are perpendicular.
, mark a point such that , where is the point of intersection of the rays and , and is the point of intersection of the rays and . Prove that the bisectors of the angles and are perpendicular.
, but this is easy by a polar duality at O.
be an arrangement of 
.
