where 
are non-negative integers that are not necessarily distinct. Find the smallest positive integer 
such that no multiple of 
be a cyclic quadrilateral. Let 
be the midpoint of the arc 
of its circumcircle which does not contain 
or 
.
and 
meet at 
and the lines 
and 
meet at 
.
and 
be a positive integer. Claire writes 
in a row on a blackboard. In a 
William can erase a number 
and replace it with either 
or 
at the same location. His goal is to perform a sequence of moves such that after he is done, the number are strictly increasing from left to right.
independent of 
such that William can always reach his goal in at most 
moves.
independent of 
moves.
Y by Adventure10
A diagonal of a convex hexagon is called long if it decomposes the hexagon into two quadrangles.
Each pair of long diagonals decomposes the hexagon into two triangles and two quadrangles.
Given is a hexagon with the property, that for each decomposition by two long diagonals the resulting triangles are both isosceles with the side of the hexagon as base.
Show that the hexagon has a circumcircle.
Each pair of long diagonals decomposes the hexagon into two triangles and two quadrangles.
Given is a hexagon with the property, that for each decomposition by two long diagonals the resulting triangles are both isosceles with the side of the hexagon as base.
Show that the hexagon has a circumcircle.
and hence 
is cyclic
