AoPS Academy
, both nowhere differentiable, such that 
is differentiable?
is a power series for which each coefficient 
is 
or 
. Show that if 
, then 
must be irrational.
![\[n!=\prod_{i=1}^n \; \text{lcm} \; \{1, 2, \ldots, \left\lfloor\frac{n}{i} \right\rfloor\}\]](http://latex.artofproblemsolving.com/3/f/0/3f0b8424645a6111de6a657a2bedf02ceced6942.png)
(Here lcm denotes the least common multiple, and 
denotes the greatest integer 
.)
, construct a sequence of random variables 
such that
Let
Find positive constants 
and 
such that
be a group and 
a nonempty subset of . Let 
. is finite, then 
if and only if 
and 
. and a nonempty subset of such that 
, 
and 
is a proper subgroup of . be a ring with unity such that for every 
there exist 
such that 
. Prove that
then 
such that 
then the result from a) may no longer hold.
are complex numbers with absolute value . For 
define 
. Given that the sequence 
converges. Show that,
.
is an integer, then there exist invertible matrices 
with non-zero entries such that:![\[A_1^{-1} + A_2^{-1} + \ldots + A_n^{-1} = (A_1 + A_2 + \ldots + A_n)^{-1}.\]](http://latex.artofproblemsolving.com/a/f/9/af9ff29771952abe37a00f3830c72e8c517e1cd6.png)
with 
right and 
, 
. On the ray 
we take two points 
and 
so that 
and 
. Consider the squares 
and 
containing inside 
and 
, respectively. If 
and 
, 
and 
, prove that 
and that , 
and 
are collinear.
with right and , . On the ray we take two points and so that and . Consider the squares and containing inside and , respectively. If and , and , prove that and that , and are collinear.
holds inequality 
topic in College Math :coolspeak:
