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)
and 
.
Equality holds when 
or 
is the incentre and point 
is defined such that 
and 
is the perpendicular bisector of 
and 
be the circles passing through 
and 
respectively such that 
to be a point such that it lies on both the circles 
and 
are perpendicular.
weights, each weighing 
and another 
weights with 
each. Write a algorithm that determines the ways in which a scale can be balanced with a weight 
on the left pan, and display the number of possible solutions. (The weights can be placed on both pans and the program starts with the numbers 
.
.
to meet 
meets the midpoint of 
.
,
.
with side length of 
.
is selected on the diagonal 
.
as 
.
.
as 
respectively, compute the area of 
given the area of 
.
.
is equilateral. Given that 
and the midpoint of 
be reals with 
.
Y by Adventure10
Let 
be a right triangle with a right angle at 
, such that 
. Let 
be the midpoint of 
and 
the intersection of 
with the perpendicular on passing through . Let 
be the intersection of the parallel to that passes through , with the perpendicular on 
passing through 
. Show that the triangles 
and 
are similar if and only if 
.
be a right triangle with a right angle at 
, such that 
. Let 
be the midpoint of 
and 
the intersection of 
with the perpendicular on passing through . Let 
be the intersection of the parallel to that passes through , with the perpendicular on 
passing through 
. Show that the triangles 
and 
are similar if and only if 
.
