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 
.
with orthocenter 
.
to meet 
.
meets the midpoint of 
.
,
.
with side length of 
.
is selected on the diagonal 
such that 
.
as 
.
.
as 
respectively, compute the area of 
given the area of 
.
.
lies inside the rectangle such that 
is equilateral. Given that 
and the midpoint of 
be reals with 
a
.
Y by Adventure10
Points are marked on the sides of the acute triangle 
so that on the side 
there are points 
between 
and 
: 
, on the 
side there are the points 
between 
and : 
, on the side of 
points 
between 
and 
: 
. Let 
be the orthocenters of triangles 
, 
, 
. Find the area of the polygon 
if 
, 
and the area of the triangle is 
.
so that on the side 
there are points 
between 
and 
: 
, on the 
side there are the points 
between 
and : 
, on the side of 
points 
between 
and 
: 
. Let 
be the orthocenters of triangles 
, 
, 
. Find the area of the polygon 
if 
, 
and the area of the triangle is 
.This post has been edited 2 times. Last edited by parmenides51, Dec 19, 2022, 12:44 PM
