AoPS Academy
as the set of all numbers of the form 
for some nonnegative 
and 
. Find (with proof) all pairs 
such that 
and 
.
over 
, and prove that your solution(s) is/are the only one(s).
be integers. Suppose that for each 
there exists an integer 
such that 
is divisible by 
. Prove that 
for some integer 
.
where 
, is cut by the line 
the tangents at the points of intersection are concurrent.
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 , the equilibrium will be stable; but if is small compared to , the equilibrium will be unstable. What is the critical value of the ratio 
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 by inches (
and are integers). He paints the entire surface of the block, cuts the block into inch cubes, and notices that exactly half the cubes are completely unpainted. Prove that the number of essentially different blocks with this property is finite. (Do not attempt to enumerate them.)
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, even. For each positive integer 
, let 
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 postive integers.
be a positive integer and let 
be an irrational number. Show that for every integer 
, there exists a positive integer such that 
.
is increasing , because 
is increasing and unbounded, and 
so take all non-negative values is an arbitrary irrational number” xD. I was confused.
is dense in 
so we split the interval into