is the incentre and point 
is defined such that 
and 
is the perpendicular bisector of 
with 
.
as 
,
and 
as 
.
.
are there such that 
and 
?
, where 
is a real number. Prove that for all positive 
then prove
and 
Prove that
Let 
Prove that
Let 
Prove that
Let 
Prove that
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 
.
Y by Adventure10, Mango247
Let 
be a point inside the acute angle 
. An arbitrary circle 
passes through 
, intersecting 
and 
at the second intersection 
and 
, respectively. Let 
be the midpoint of 
. Prove that is always on a fixed line (when changes, but always goes through 
and ).
be a point inside the acute angle 
. An arbitrary circle 
passes through 
, intersecting 
and 
at the second intersection 
and 
, respectively. Let 
be the midpoint of 
. Prove that is always on a fixed line (when changes, but always goes through 
and ).
be the perpendicular bisector of 
as a line, and let 
.
be the center of 
be the midpoints of 
respectively. Note that 
varies linearly with 
by vectors; similarly, since 
varies linearly with 
varies linearly with 
and the line 
,
.
with 
and 
,![$y=\frac{\sin s [2b\cos s \cdot x-(2ax-a^{2}-b^{2})\sin s]}{2(b\cos^{2}s-a\sin s\cos s+b)}$](http://latex.artofproblemsolving.com/e/9/5/e956bd0d4a0bddc13f4921d464466f212932cede.png)
.
