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 
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 
.
Y by Adventure10
On the side 
of the triangle 
the point 
is taken in such a way that the perpendicular bisector of segment 
passes through the center of the circle inscribed in the triangle . Prove that this perpendicular passes through a vertex of the triangle 
.
of the triangle 
the point 
is taken in such a way that the perpendicular bisector of segment 
passes through the center of the circle inscribed in the triangle . Prove that this perpendicular passes through a vertex of the triangle 
.This post has been edited 2 times. Last edited by parmenides51, Mar 25, 2021, 7:24 AM
Reason: vertex instead of vertice
Reason: vertex instead of vertice
.
.
on 
,
