be a positive integer that is not divisible by 
.
be a positive integer. Prove that none of the numbers of the form 
is a square number.
different bus lines in a city, each stopping at exactly 
stations and running in both directions. Nevertheless, for every two different stations there is always a bus line connecting these two stations. Determine the maximum number of stations in this city.
be an isosceles triangle with 
and circumcircle 
.
perpendicular to 
is denoted by 
.
be any point on 
with center 
intersects 
once more at point 
and the circumcircle 
.
and 
lie on a straight line.
be a positive integer. Furthermore, let ![$x_1, x_2,\ldots, x_n \in [0, 2]$](http://latex.artofproblemsolving.com/b/2/a/b2a6731961601353e26224c285a32923fbb54eb3.png)
be real numbers subject to 
.
When does equality hold?
be an odd integer greater than 
Let 
be an 
symmetric matrix such that each row and column consists of some permutation of the integers 
Show that each of the integers 
must appear in the main diagonal of 
convergent?
and radius 
and the base of a hemisphere of radius 
This body is placed with the hemispherical end on a horizontal table, with the axis of the cylinder in a vertical position, and then slightly oscillated. It is intuitively evident that if 
is large as compared to 
which enables the body to rest in neutral equilibrium in any position?
![\[ \begin{vmatrix} 0 & a_1 - x & a_2 - x \\ -a_1 - x & 0 & a_3 - x \\ -a_2 - x & -a_3 - x & 0\end{vmatrix} = 0 \qquad (a_i \neq 0) \]](http://latex.artofproblemsolving.com/7/2/1/721620c769b5622dc4172dd38b7a924b5d95e694.png)
shall have a multiple root.
by 
South Latitude. The parallels of latitude are concentric circles with radii proportional to their co-latitudes. Australia is near the periphery of the map and is intersected by the parallel of latitude 
S.In the very close vicinity of this parallel how much are East and West distances exaggerated as compared to North and South distances?
and 
be relatively prime positive integers, 
,
be chosen so that
is a minimum. Prove that
be a permutation of the integers 
Call 
a big integer if 
for all 
Find the mean number of big integers over all permutations on the first 
be two 
be a positive integer. Prove that 
if and only if 
.
Y by kiyoras_2001
Kuba has two finite families 
of convex polygons (in the plane). It turns out that every point of the plane lies in the same number of elements of 
as elements of 
. Prove that 
.
\textit{Note:} We treat segments and points as degenerate convex polygons, and they can be elements of or .
of convex polygons (in the plane). It turns out that every point of the plane lies in the same number of elements of 
as elements of 
. Prove that 
.\textit{Note:} We treat segments and points as degenerate convex polygons, and they can be elements of
or .
