and natural number 
,
are there such that 
and 
?
and 
.
be real numbers such that 
Prove that
Let 
Prove that
and 
, then we can deduce 
\le 2 $ ?
satisfying 
.![\[
0 \le (x^2 + 1)(y^2 + 1) + 4(x - 1)(y - 1) \le 164.
\]](http://latex.artofproblemsolving.com/4/2/d/42d0b88388462f16dd99d928b94024e689ba7c2e.png)
be a sequence defined by 
.
if 
.
that satisfy 
with 
,
and area 
.
with diameter 
meets 
at 
.
,
and extend to meet 
such that 
and 
is the diameter of 
of 
,
of 
Y by Adventure10
In the below figure, there is a regular hexagon and three squares whose sides are equal to 
cm. Let 
, and 
be the centers of the squares. The perimeter of the triangle 
can be written in the form 
(cm), where 
are integers. Compute the value of 
.
cm. Let 
, and 
be the centers of the squares. The perimeter of the triangle 
can be written in the form 
(cm), where 
are integers. Compute the value of 
.This post has been edited 1 time. Last edited by parmenides51, Jan 31, 2020, 10:18 AM
is equilateral and its sides are parallel to the hexagon. Consider the three mini triangles that are inside the squares. It is easy to see that they are all equilateral. Since the height of one of these mini triangles is 
their side length is 
Thus, the side length of 
The triangle's perimeter is then 
The requested sum is 
