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 
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 
.
with orthocenter 
.
to meet 
.
meets the midpoint of 
,
.
.
is selected on the diagonal 
such that 
.
as 
.
.
as 
respectively, compute the area of 
given the area of 
.
.
is equilateral. Given that 
and the midpoint of 
be reals with 
.
Y by Amir Hossein, Adventure10, Mango247
Let 
be a convex quadrilateral whose sides 
and 
are not parallel. Suppose that the circles with diameters 
and 
meet at points 
and 
inside the quadrilateral. Let 
be the circle through the feet of the perpendiculars from to the lines 
, and . Let 
be the circle through the feet of the perpendiculars from to the lines 
, and . Prove that the midpoint of the segment 
lies on the line through the two intersection points of and .
be a convex quadrilateral whose sides 
and 
are not parallel. Suppose that the circles with diameters 
and 
meet at points 
and 
inside the quadrilateral. Let 
be the circle through the feet of the perpendiculars from to the lines 
, and . Let 
be the circle through the feet of the perpendiculars from to the lines 
, and . Prove that the midpoint of the segment 
lies on the line through the two intersection points of and .
