[/quote]
,
and 
are natural numbers such that
is a prime number. Find all possible values of 
.
be a positive integer. Aslı and Zehra are playing a game on an 
grid. Initially, 
stones are placed on some of the unit squares of this grid.
![\[\frac{x^3+1}{(y-1)(z+1)}+\frac{y^3+1}{(z-1)(x+1)}+\frac{z^3+1}{(x-1)(y+1)}\]](http://latex.artofproblemsolving.com/5/c/6/5c6894b617b990ec69c62ed1cf7d062d4660af53.png)
where 
are reals.
and let 
be positive integers such that 
.
and determine when equality holds.
equals to a positive integer power of 
.
![\[\frac{\sin^2A+\sin^2B+\sin^2C}{\cos^2A+\cos^2B+\cos^2C} = 2\]](http://latex.artofproblemsolving.com/9/2/a/92a524bbdca92e9dd4091275714bf346bad96cd7.png)
is a pentagon whose vertices lie on circle 
where 
.
and 
intersect at 
,
meet 
at 
.
is the midpoint of arc 
on 
holds, then what is the value of 
?
be a cyclic quadrilateral and let the intersection of 
.
be arbitrary points on sides 
and 
.
.
satisfy the equality, 
prove that the triangle must have a right angle.
is an odd integer. Each candidate is evaluated by each judge as either pass or fail. Suppose that each pair of judges agrees on at most 
candidates. Prove that ![\[{\frac{k}{m}} \geq {\frac{n-1}{2n}}. \]](http://latex.artofproblemsolving.com/3/8/7/387c8824b28f6422306e3c25b34fccb0a841f4cd.png)
be real numbers such that![\[ a^2+x^2=b^2+y^2=c^2+z^2=(a+b)^2+(x+y)^2=(b+c)^2+(y+z)^2=(c+a)^2+(z+x)^2 \]](http://latex.artofproblemsolving.com/7/e/b/7ebfb2f663d390714310d3c43b9c68e1b101d632.png)
.
,
and 
,
and radius 
intersects 
is tangent to the circles with diameters 
and 
.
of non-negative real numbers such that the following three conditions are all satisfied:
;
;
of 
[/list]
is a permutation of 
,
.
for which there exist real numbers 
satisfying 
,
and
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 
.
)
,
.
.
