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such that
be a positive integer larger than 
. Build an infinite set 
of subsets of 
having the following properties: distinct sets of have exactly one common element;
distinct sets of have void intersection.
and 
be positive integers such that 
divides 
. Prove that 
.
![$f\in C^1[a,b]$](http://latex.artofproblemsolving.com/8/4/8/848da12534d564560fe1cb016142159234be98b4.png)
, 
and suppose that 
, 
, is such that
for all ![$x\in [a,b]$](http://latex.artofproblemsolving.com/7/8/d/78d9dcd969a2c45f7cc9234aa1377b6ec1c2d5d3.png)
. Is it true that 
for all ?
![$f \in C([0,1];[0,1])$](http://latex.artofproblemsolving.com/4/7/5/475518b7db0b787dd90b93f602c7b912c663fba3.png)
bijective, 
and ![$(a_k) \in [0,1]^\mathbb N$](http://latex.artofproblemsolving.com/a/d/8/ad8a80d847c60ee8e916a1f9b08b7b03ec17c814.png)
with 
converge.
converge?
is called perfect if the sum of its positive divisors 
is twice , that is, 
. For example, 
is a perfect number since the sum of its positive divisors is 
, which is twice . Prove that if is a positive perfect integer, then:![\[
\sum_{p|n} \frac{1}{p + 1} < \ln 2 < \sum_{p|n} \frac{1}{p - 1}
\]](http://latex.artofproblemsolving.com/0/c/0/0c08f527e87f7aae4ca5226f19bd084249551fe3.png)
where the sums are taken over all prime divisors 
of .
![$f\in C[0,b]$](http://latex.artofproblemsolving.com/8/9/8/898aec0300888e562794b7a551be34f5c2676236.png)
, 
and let 
be periodic with period . Prove that 
has a limit as 
and
be an even positive integer. Let 
be a monic, real polynomial of degree 
; that is to say, 
for some real coefficients 
. Suppose that 
for all integers such that 
. Find all other real numbers 
for which 
.
where 
, 
, and 
are nonnegative integers.
, 
, 
and 
for 
. Prove that 
and give an example where 
.
on 
for which![\[
\lim_{R \to \infty} \int_{\|x\|_2 \leq R} f(x) \, dx = 0
\]](http://latex.artofproblemsolving.com/4/6/9/469f69ee93ae8d1de73c31ceb066253751aa9775.png)
and for which![\[
\lim_{R \to \infty} \int_{\|x\|_\infty \leq R} f(x) \, dx \text{ does not exist.}
\]](http://latex.artofproblemsolving.com/d/f/a/dfadbdd902375b9f67cd2dd0dbb494978bd9e75c.png)
be the set of all 
matrices with at most two entries in each row equal to and all other entries equal to 
.
.
denotes the determinant of the matrix .
are on the side 
such that 
and let 
be the projections of 
onto 
respectively. The line 
intersects 
at 
(different from point ).
.
.
, it's easy to know that 
, from which we can get 
.
, which is the area of 
, as desired.
. We have![\begin{align*}
2[AMDN] &= 2([AMD]+[AND]) \\
&= AD(AM \sin \theta + AN \sin(A-\theta)) \\
&= AD(AF \sin \theta \cos(A-\theta) + \cos \theta \sin(A-\theta)) \\
&= AD \cdot AF \cdot \sin \theta.
\end{align*}](http://latex.artofproblemsolving.com/9/a/f/9afd7bdeab076d2d8de1ddae51b2a174a41c222e.png)
So it suffices to show that 
, but this is simply true because .
Prove that
Let 
Prove that![$$ (a+b)(a+1)(b+1) \leq \frac{3}{2}+\sqrt[3]{6}+\sqrt[3]{36}$$](http://latex.artofproblemsolving.com/f/5/e/f5e68204507aa9c3e71c28844bbee3a804546b94.png)
Now dividing both sides by 
gives the result.
Divide by 
and use 
.
Using Lemma 1: 
This is equivalent to 
Which is just Ptolemy theorem for 
.![[asy]
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clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);
[/asy]](http://latex.artofproblemsolving.com/a/0/b/a0b5111ee09630046735ad62d794f07764d2fd5a.png)
what is it?
.
is perpendicular to 
，
,
,
,
