& 
.
with 
and at most one of them being zero, show that![\[\frac{1}{\max(a^2,b)}+\frac{1}{\max(b^2,c)}+\frac{1}{\max(c^2,a)}\geq\frac{27}{4}\]](http://latex.artofproblemsolving.com/9/e/1/9e14a9ef2d1a0a98006bcf716fa33c3d9ea29bd5.png)
be an integer. Find all sequences 
satisfying the following conditions:![\[ \text{ (a) } a_i \in \left\{0,1\right\} \text{ for all } 1 \leq i \leq n^2 + n;
\]](http://latex.artofproblemsolving.com/3/c/5/3c509ec2e9013e8d3be492c8eb44a7c33841b74e.png)
![\[ \text{ (b) } a_{i + 1} + a_{i + 2} + \ldots + a_{i + n} < a_{i + n + 1} + a_{i + n + 2} + \ldots + a_{i + 2n} \text{ for all } 0 \leq i \leq n^2 - n.
\]](http://latex.artofproblemsolving.com/9/7/d/97d2a467d1c0dc8594ec024c3bb9b8c87ee85b19.png)
Y by Adventure10, Mango247
The base of a regular quadrilateral pyramid 
is a square with side length 
and its lateral edge has length a
. Let 
be a point inside the pyramid. Consider the five pyramids which are similar to , whose top vertex is at and whose bases lie in the planes of the faces of . Show that the sum of the surface areas of these five pyramids is greater or equal to one fifth the surface of , and find for which equality holds.
is a square with side length 
and its lateral edge has length a
. Let 
be a point inside the pyramid. Consider the five pyramids which are similar to , whose top vertex is at and whose bases lie in the planes of the faces of . Show that the sum of the surface areas of these five pyramids is greater or equal to one fifth the surface of , and find for which equality holds.
