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 
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 
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 
.
that satisfy 
.
to meet 
meets the midpoint of 
,
.
Y by
Let 
the midpoint of 
of an acutangle triangle 
with 
. Let 
the circumcircle of . Let 
the intersection of the tangents to in point 
and 
and 
the intersection of 
and . Let 
the altitude of triangle 
with 
in and 
the circumcircle of triangle 
. Let 
and the intersections of and 
. Prove that 
.
the midpoint of 
of an acutangle triangle 
with 
. Let 
the circumcircle of . Let 
the intersection of the tangents to in point 
and 
and 
the intersection of 
and . Let 
the altitude of triangle 
with 
in and 
the circumcircle of triangle 
. Let 
and the intersections of and 
. Prove that 
.
is cyclic.
is tangent to 
and as 
with 
.
,
hence 
is cyclic, so 
is cyclic. We now have
hence
