'is a perfect square.
Prove that
and 
.
Let 
.
and 
Prove that
and 
be positive real numbers with 
.
For which 
such that 
for all 
,
.
and 
of an equilateral triangle 
are taken points 
and 
,
, points 
and 
so that 
.
and 
is equal to 
.
over 
.
![\[\lfloor x \rfloor\]](http://latex.artofproblemsolving.com/d/3/f/d3f8facf62413e1f17c89b9c183a8735c2defe8b.png)
denote the greatest integer less than or equal to x . What is the sum of the squares of the real numbers x for which ![\[ x^2 - 20\lfloor x \rfloor + 19 = 0 \]](http://latex.artofproblemsolving.com/f/a/9/fa96f2d6217a16029e5f03499e0a83deffdfa28a.png)
Y by
It is known that among all 
-sided polygons inscribed in a circle, the regular n-gon has the greatest perimeter. Using this fact, prove that:
(i) if four points
are chosen on a circle such that the sum of all six distances 
is maximal then the points are vertices of a square.
(ii) if six points
are chosen on a circle such that the sum of all fifteen distances 
is maximal then the points are vertices of a regular hexagon.
-sided polygons inscribed in a circle, the regular n-gon has the greatest perimeter. Using this fact, prove that:(i) if four points
are chosen on a circle such that the sum of all six distances 
is maximal then the points are vertices of a square.(ii) if six points
are chosen on a circle such that the sum of all fifteen distances 
is maximal then the points are vertices of a regular hexagon.This post has been edited 2 times. Last edited by parmenides51, Jan 23, 2021, 11:33 PM
Reason: title typo
Reason: title typo
