.
and 
,
is not less than the area of 
.
be an acute triangle. Denote by 
the foot of the perpendicular line drawn from the point 
to the side 
,
the midpoint of 
the orthocenter of 
be the point of intersection of the circumcircle 
of the triangle 
,
be the point of intersection (other than 
)
and the circle 
must hold.
the length of the line segment 
such that 
(where 
)
that satisfies 
for all 
is the function obtained by composing 
)
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
Of a regular polygon of 
sides, there are 
diagonals which pass through the centre of the inscribed circle. The angles which these diagonals subtend at two given points 
and 
on the circumference, are 
and 
. Prove that 
.
sides, there are 
diagonals which pass through the centre of the inscribed circle. The angles which these diagonals subtend at two given points 
and 
on the circumference, are 
and 
. Prove that 
.This post has been edited 1 time. Last edited by parmenides51, Apr 22, 2021, 7:41 PM
