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 
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
On the hypotenuse 
of a right-angled triangle 
in the outer side is a congruent triangle 
, 
, 
(see fig.). The circle 
, inscribed in a triangle , touches the hypotenuse 
at point 
, and the circle 
inscribed in triangle , touches leg 
at point 
. Prove that the segment 
, the hypotenuse and the segment connecting the centers of the circles and , intersect at one point.
of a right-angled triangle 
in the outer side is a congruent triangle 
, 
, 
(see fig.). The circle 
, inscribed in a triangle , touches the hypotenuse 
at point 
, and the circle 
inscribed in triangle , touches leg 
at point 
. Prove that the segment 
, the hypotenuse and the segment connecting the centers of the circles and , intersect at one point.
This post has been edited 1 time. Last edited by parmenides51, Nov 17, 2020, 10:27 AM
