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 
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 Adventure10, Mango247
Two circles 
and 
touch externally at 
. their external common tangent touches at A and at 
. A line, parallel to 
and tangent to at 
, intersects at 
and 
. Prove that the common chord of the circumcircles of triangles 
and 
passes through point .
(A. Kalinin)
and 
touch externally at 
. their external common tangent touches at A and at 
. A line, parallel to 
and tangent to at 
, intersects at 
and 
. Prove that the common chord of the circumcircles of triangles 
and 
passes through point .(A. Kalinin)
This post has been edited 1 time. Last edited by parmenides51, Jul 29, 2018, 7:37 PM
Reason: source typo
Reason: source typo
lies on 
,
is 
is orthogonal to 
,
