or 
.
and 
and 
operated by two different companies, then there must exist flight route 
operated by the third company different from 
,
and 
denote the number of flight routes operated by companies 
.
is a positive integer. Determine all functions 
such that![\[f(x^{n+1}+y^{n+1})=x^nf(x)+y^nf(y),\forall x,y>1.\]](http://latex.artofproblemsolving.com/a/4/7/a47d53806f2148cc3339335fb990b33641085896.png)
be a fixed integer. Find the least constant ![\[\sum_{i<j} x_{i}x_{j} \left(x^{2}_{i}+x^{2}_{j} \right) \leq C
\left(\sum_{i}x_{i} \right)^4\]](http://latex.artofproblemsolving.com/d/d/e/dde2d1dd0c412d8ba640f1d4c044e2fa2dcccc82.png)
(the sum on the left consists of 
summands). For this constant 
,
be the least prime that does not divide 
,
to be the product of all primes less than 
For 
,
.
defined by 
and ![\[ x_{n+1} = \frac{x_n p(x_n)}{q(x_n)} \]](http://latex.artofproblemsolving.com/8/2/5/825be3bbef6a29de69aead03392ca7215093f320.png)
for 
.
.
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 
.
)
,
.
.
