be a positive integer. Claire writes 
distinct positive real numbers 
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.
a friendly number if there exists a positive rational number 
such that 
.
such that 
is on the same side of 
as 
and 
.
intersects 
at 
.
on 
such that 
is equilateral.
Y by Adventure10, Mango247
As shown in the figure, there are two rectangles 
and 
with the same area, and with parallel corresponding edges. Let points 
and 
be the midpoints of segments 
and 
, respectively. Prove that points 
and lie on the same line.
Proposed by Morteza Saghafian
and 
with the same area, and with parallel corresponding edges. Let points 
and 
be the midpoints of segments 
and 
, respectively. Prove that points 
and lie on the same line.
Proposed by Morteza Saghafian
now by simple similiarty prove 
is the midpoint of 
so 
and we are done 
.
and 
which implies 
.
and 
.
is a parallelogram. 
and 
is the median of 
which means 
is the median of 
.
are colinear.
are colinear. But 
are colinear. Hence, proved.
