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 
with orthocenter 
.
to meet 
meets the midpoint of 
,
.
Y by
Let 
be a circle centered at 
. Let 
be the chord of this circle, 
be the midpoint of this chord. Tangents to the circle at points 
and 
intersect at point 
. The line 
passes through the point , intersects the shorter arc at the point 
and the longer arc at the point 
so that 
. Prove that the center of the circumcircle of 
is a point symmetric to the point wrt line 
.
be a circle centered at 
. Let 
be the chord of this circle, 
be the midpoint of this chord. Tangents to the circle at points 
and 
intersect at point 
. The line 
passes through the point , intersects the shorter arc at the point 
and the longer arc at the point 
so that 
. Prove that the center of the circumcircle of 
is a point symmetric to the point wrt line 
.This post has been edited 1 time. Last edited by parmenides51, Jul 15, 2021, 11:38 AM
