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 
such that 
.
as 
.
.
as 
respectively, compute the area of 
given the area of 
.
.
lies inside the rectangle such that 
is equilateral. Given that 
and the midpoint of 
be reals with 
.
Y by jhu08
Let 
be the intersection point of the altitudes 
, 
and 
of an acute triangle 
and let 
, 
, 
be the midpoints of the sides 
, 
and 
respectively. Suppose that does not lie into the interior of triangle 
. Show that at least one of the areas of triangles 
, 
, and 
is less than 
of the area of triangle .
be the intersection point of the altitudes 
, 
and 
of an acute triangle 
and let 
, 
, 
be the midpoints of the sides 
, 
and 
respectively. Suppose that does not lie into the interior of triangle 
. Show that at least one of the areas of triangles 
, 
, and 
is less than 
of the area of triangle .This post has been edited 4 times. Last edited by parmenides51, Dec 15, 2022, 7:05 PM
