[/quote]
,
be a positive integer. Define a sequence by setting 
and, for each 
,
be the unique integer in the range 
for which 
is divisible by 
.
the obtained sequence is 
.
eventually becomes constant.
,
lie on sides 
,
.
meet at orthocenter 
.
and 
lie on segment 
such that 
and 
.
and 
intersect at point 
.
.
is the midpoint of side 
And 
.
over 
intersects lines 
and 
and 
and 
.
lies on 
.
such that 
prove that the triangle must have a right angle.
such that for any real numbers 
and 
,![\[
f\left(x^2+2y f(x)\right) + (f(y))^2 \leq (f(x+y))^2
\]](http://latex.artofproblemsolving.com/8/4/3/8437449cd3f8f0da24d17a60b55dcb563e6b5655.png)
defined by 
and![\[
u_{n+1} = \frac{1}{2}u_n^2 - 4 \quad \text{for all } n \in \mathbb{N}.
\]](http://latex.artofproblemsolving.com/9/9/4/994aa754cc1288ce4f28a95a0276e64282fb5f66.png)
a) Prove that there exist infinitely many positive integers 
such that 
.![\[
\lim_{n \to \infty} \frac{2u_{n+1}}{u_0u_1\cdots u_n}.
\]](http://latex.artofproblemsolving.com/f/f/1/ff174e7431cbfbc17c650d109651241286756a1a.png)
is a cyclic quadrilateral. Let 
and 
.
and 
the orthocenters of triangles 
and 
,
,
be a triangle. A circle passing through 
and 
intersects the sides 
and 
again at 
and 
respectively. Prove that 
,
and 
are concurrent, where 
and 
are the orthocentres of triangles 
respectively.
.
and 
=
.
is the foot of the altitude from 
,
,
+ 
+ 
= 
,
,
+ 
+
= 0 holds
= 
= 
+ 
+ 
.
Y by Adventure10, Adventure10, Mango247
Let 
be a convex polygon in a plane, 
its perimeter and 
its area. Let 
be the locus of all points in the space whose distance to is 
and 
is the volume of the solid 
a.) Prove that![\[V (R) = \frac 43 \pi R^3 +\frac{\pi}{2} lR^2 +2SR.\]](//latex.artofproblemsolving.com/c/7/7/c771c7c19b35c8ea5ec375184553b5550f582149.png)
Hereby, we say that the distance of a point
to a figure is 
if there exists a point 
of the figure such that the distance 
is 
(This point may lie on the boundary of the figure and inside the figure.)
additional question:
b.) Find the area of the planar
-neighborhood of a convex or non-convex polygon 
c.) Find the volume of the-neighborhood of a convex polyhedron, e. g. of a cube or of a tetrahedron.
Note by Darij: I guess that the ''-neighborhood'' of a figure is defined as the locus of all points whose distance to the figure is
be a convex polygon in a plane, 
its perimeter and 
its area. Let 
be the locus of all points in the space whose distance to is 
and 
is the volume of the solid 
a.) Prove that
![\[V (R) = \frac 43 \pi R^3 +\frac{\pi}{2} lR^2 +2SR.\]](http://latex.artofproblemsolving.com/c/7/7/c771c7c19b35c8ea5ec375184553b5550f582149.png)
Hereby, we say that the distance of a point
to a figure is 
if there exists a point 
of the figure such that the distance 
is 
(This point may lie on the boundary of the figure and inside the figure.)additional question:
b.) Find the area of the planar
-neighborhood of a convex or non-convex polygon 
c.) Find the volume of the
-neighborhood of a convex polyhedron, e. g. of a cube or of a tetrahedron.Note by Darij: I guess that the ''
-neighborhood'' of a figure is defined as the locus of all points whose distance to the figure is This post has been edited 2 times. Last edited by orl, Sep 2, 2004, 12:10 PM
"looks like". If we place spheres of radius 
(this has volume 
)
,
are the sides of the polygon), and whatever remains around the vertices. This last part is made up of sphere slices, which we can slide around on the sides of the polygon so that they form a sphere of radius 
.
)
,
.
.
