AoPS Academy
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 such that Claire can choose the initial numbers such that William cannot attain his goal in less than 
moves.
a friendly number if there exists a positive rational number 
such that 
. Find, with proof, the second smallest friendly number.
, we construct equilateral triangle 
such that 
is on the same side of 
as 
and 
. It is given that 
intersects 
at 
. Prove that there exists a point 
on 
such that 
is equilateral.
be an equilateral triangle. On the plane 
rise the perpendiculars 
and 
on the same side of the plane, so that 
and 
. Determine the measure the angle between the planes and 
.
be an equilateral triangle. On the plane rise the perpendiculars and on the same side of the plane, so that and . Determine the measure the angle between the planes and .
.Then 
so 
. Thus the plane 
is perpendicular to 
and the angle between 
and is equal to 
.
