be a function which is continuous on ![\( [0,1] \)](http://latex.artofproblemsolving.com/0/1/0/01026be294289be8f090c7b2586dde64a0d5b479.png)
and differentiable on 
,
.
such that 
.
such that 
.
be a positive integer. Prove that 
is a permutation polynomial mod 
a permutation polynomial mod 
is bijective.
such that 
?
.
as the average of all 
for 
.![\[ a = \lim_{k \to \infty} \sup_{n \geq k} \frac{a_n}{n^4} .\]](http://latex.artofproblemsolving.com/5/2/d/52d93af7aafa6657689246d9a74c8eb05d96efa3.png)
with 
such that
and 
and 
with equal radii intersect at P and Q. Points B and C are located on the circle
The intersection point of the circumcircles of triangles XPC and YPB is called S. Prove that BC, XY and QS are concurrent.
? (there is an easy workaround)
what is range of values of 
what is range of values of 
be a sequence of nonnegative real numbers whose sum is 
.![$$\sum_{k=1}^\infty \left(k\cdot (3k+2) \cdot 2^{-k} \cdot \sqrt[k]{a_1a_2\dots a_k}\right).$$](http://latex.artofproblemsolving.com/2/d/f/2df5179ce532077733530d5e9f1266c22d1cdb7d.png)
Y by
Does the math many people on AOPS work for really make a large impact? I get that this is a place for math but I think I'm losing touch with my prior math skill. AMC and AIME are solving problems that have answers and just take practice but math as a job or just like an accountant doesn't need/needs a lot more than this type of math. I might switch to another stem category for college apps pathway. Also, many people who are already very good at competition math don't get good offers because there's people who are even better(note: i get the point of math isn't only college). I think if I work *hard* I'll be AIME qual but I think many people would have higher achievements with the same amount of work in another category. I think competition math really just trains for problem solving, unique thinking, and pattern recognition(aka IQ pts).
