, where 
is a real number. Prove that for all positive 
then prove
and 
Prove that
Let 
Prove that
Let 
Prove that
Let 
Prove that
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 
is tangent to 
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 
.
with orthocenter 
.
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 
.
Y by
A circle with radius 
and a circle with radius 
have the same center. The area of the small circle we call 
. The area of the ring bounded by both circles we call 
. The triangle 
is right. A straight line cuts the hypotenuse ![$[V W]$](//latex.artofproblemsolving.com/4/f/a/4fa109038266e0fa7dbb9c20c644548b7e57fb91.png)
perpendicularly 
and cuts ![$[UW]$](//latex.artofproblemsolving.com/e/d/9/ed9d18cd1e42db3bf822bc21c30e8ded0d07edf8.png)
into 
. Take 
such that 
. The area of 
we call 
and the area of the quadrilateral 
we call 
. Prove that 
if and only if 
.
and a circle with radius 
have the same center. The area of the small circle we call 
. The area of the ring bounded by both circles we call 
. The triangle 
is right. A straight line cuts the hypotenuse ![$[V W]$](http://latex.artofproblemsolving.com/4/f/a/4fa109038266e0fa7dbb9c20c644548b7e57fb91.png)
perpendicularly 
and cuts ![$[UW]$](http://latex.artofproblemsolving.com/e/d/9/ed9d18cd1e42db3bf822bc21c30e8ded0d07edf8.png)
into 
. Take 
such that 
. The area of 
we call 
and the area of the quadrilateral 
we call 
. Prove that 
if and only if 
.This post has been edited 4 times. Last edited by parmenides51, Dec 23, 2022, 7:23 PM
