be a point in the plane, distinct from the vertices of 
.
the reflections of 
and 
,
are collinear if and only if 
coincide, prove that 
such that ![\[f(2x+y) + f(x+f(2y)) = f(x)f(y) - xy\]](http://latex.artofproblemsolving.com/0/6/2/0623b7c328af6d088cfc5dd288afe3e17a94ff29.png)
for all reals 
and 
.
be a fixed positive integer, and 
be a sequence of nonnegative real numbers such that, for all 
,
.
is bounded above.
is bounded above.
are positiv real number, such that 
prove inequality 
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 
.
.
to meet 
.
meets the midpoint of 
.
,
.
with side length of 
.
is selected on the diagonal 
.
as 
.
.
as 
respectively, compute the area of 
given the area of 
.
Y by
Point 
is selected on the median 
of triangle 
. Line 
intersects side 
at point 
, and line 
intersects side 
at point 
. It turned out that the quadrilateral 
is cycic. Prove that triangle is isosceles.
is selected on the median 
of triangle 
. Line 
intersects side 
at point 
, and line 
intersects side 
at point 
. It turned out that the quadrilateral 
is cycic. Prove that triangle is isosceles.This post has been edited 1 time. Last edited by parmenides51, Jan 3, 2021, 10:33 AM
Reason: latex
Reason: latex
and using 
we get 
,
is a harmonic bundle. If we take perspectivity from 
gives harmonic bundle 
such that 
.
is also a harmonic bundle, where 
is the foot of the altitude from 
to 
,
are isogonal 
bisects 
,
