be a set of Natural numbers from 
to 
.
)
.
and 
,
is not less than the area of 
,
inequalities, these given comparisons satisfy the following three constraints:
and 
,
and 
respectively, then the sum of elements of the union 
is greater than that of 
.
that satisfies all these conditions?
then prove \frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a}+\frac{1}{2}\cdot\frac{abc}{ab+bc+ca}\ge \frac{5}{3}.$$
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 
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 
.
that satisfy 
.
to meet 
.
meets the midpoint of 
.
,
.
Y by
Triangle 
is given. The line perpendicular to 
and going and through 
intersects the circle with diameter at the points 
and 
where lies closer to than . Analogously the line perpendicular to 
and going through 
intersects the circle with diameter at the points 
and 
where lies closer to than . Prove that the intersection point of 
and 
lies on the line 
.
is given. The line perpendicular to 
and going and through 
intersects the circle with diameter at the points 
and 
where lies closer to than . Analogously the line perpendicular to 
and going through 
intersects the circle with diameter at the points 
and 
where lies closer to than . Prove that the intersection point of 
and 
lies on the line 
.
,
.
be the foot of altitude from 
is cyclic.
and 
.
lies on 
and 
lies on 
.
is the radical axis of 
and 
and 
is the radical axis of 
and 
and 
are 
;
is cyclic by easy angle chase,
Now we are able to show the collinearity,
We are done.
