[/quote]
,
so that:
with all real numbers 
satisfies 
be a positive integer. Do there exist distinct 
and 
,
or 
,
divides 
?
be a triangle and 
be the midpoints of 
respectively. Let 
and 
be points in plane such that 
.
pass through a fixed point.
such that:
such that:
such that for any 
,
such that 
.
of the acute-angled triangle 
is tangent to its side 
at a point 
.
be an altitude of triangle 
be the midpoint of the segment 
is the common point of the circle 
(distinct from 
are tangent to each other at the point 
is not a perfect cube of an integer.
over 
is linear
such that for all 
,![\[f(x+yf(x))+y = xy + f(x+y).\]](http://latex.artofproblemsolving.com/c/f/3/cf3d20a041c27244e90876119b4568b7a3e13c03.png)
be a cyclic pentagon with 
and 
.
be points on 
and 
,
is cyclic. Let 
be the midpoint of 
.
,
,
concur.
be real numbers such that 
and 
and determine the equality cases.
be the set of real numbers. Determine all functions ![\[f(x+f(x+y))+f(xy)=x+f(x+y)+yf(x)\]](http://latex.artofproblemsolving.com/6/b/2/6b2acc0a1d76866e461fdf5d96126806a9ead8dd.png)
for all real numbers 
with incenter 
.
and 
intersects 
and 
at 
and 
,
be the midpoints of 
meets the external bisector of angle 
at 
and line 
meets the external bisector of angle 
at 
.
are concylic.
Y by Adventure10, Mango247
There is a convex polyhedron with 
faces.
Show that if more than
of the faces are such that no two have a common edge,
then the polyhedron cannot have an inscribed sphere.
faces.Show that if more than
of the faces are such that no two have a common edge,then the polyhedron cannot have an inscribed sphere.
This post has been edited 1 time. Last edited by parmenides51, Feb 19, 2020, 7:16 AM
Reason: typo: added ''more than'', see #3
Reason: typo: added ''more than'', see #3
