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, 
such that 
.
. For 
, define 
recursively by![\[
f_n(x) = 2\pi \sin(f_{n-1}(x)).
\]](http://latex.artofproblemsolving.com/1/3/6/1362dadbe22dd49aa7f8bf96c9e8f0d9ec73ffe1.png)
How many intervals ![$[a, b]$](http://latex.artofproblemsolving.com/d/a/2/da2e551d2ca2155b8d8f4935d2e9757722c9bab6.png)
are there such that
, 
, 
, and 
is increasing on ?
be a bijection. Prove or disprove the following claims:
for each such 
[/*]
for each such 
[/*]
, the equation 
is always solvable. One can see that for the prime 
, 
are the solutions. For case we see that the equation becomes 
and this doesn't have any solutions. The case 
can be handled in 
cases. The first case where 
and 
. From the well known facts that 
and 
are quadratic residues for primes of the form 
we see that the congruence 
is solvable (of course with 
). What to do for the case 
? Should we consider the sets 
and 
and show that they intersect? I haven't tried this line of thought though, but it would nice to have an argument which works specifically for the case 
.
, there exists a positive integer 
such that the following holds:
, there exist a positive integer 
and 
positive integers 
such that for all integers 
,![\[\sum_{i=1}^s x_i^j = \sum_{i=1}^s y_i^j,\]](http://latex.artofproblemsolving.com/5/a/8/5a821bfebe63aa8791e7498b1ba97654b372ea3e.png)
but![\[\sum_{i=1}^s x_i^{k+1} \neq \sum_{i=1}^s y_i^{k+1}.\]](http://latex.artofproblemsolving.com/c/9/3/c931dc97085a884c57c2b2c8439b9bc82cc80141.png)
Proposed by Mucong Sun, Tsinghua University
such that there exist complex-coefficient polynomials 
satisfying![\[\{z \in \mathbb{C}: |\operatorname{Re}(z)|^r + |\operatorname{Im}(z)|^r = 1\}[= \left\{ \frac{P(w)}{Q(w)} : w \in \mathbb{C}, |w| = 1 \text{ and } Q(w) \neq 0 \right\}.\]](http://latex.artofproblemsolving.com/d/3/7/d370098a6116115a74d815472661baf277a7e341.png)
Created by Cheng Jiang, Massachusetts Institute of Technology
. The spectral radius of 
, denoted by 
, is defined as![\[
\rho(A) = \max_i |\lambda_i|
\]](http://latex.artofproblemsolving.com/1/9/6/19680614e63b45f96087d0c0f8360cf1029b19d0.png)
where 
are all the eigenvalues of the matrix .. There exists a norm 
such that 
if and only if the spectral radius of satisfies the condition 
.
be a differentiable function. If 
, 
and 
for all 
. Which of the followings are necessarily true:
f is strictly increasing
![\[
\lim_{n \rightarrow \infty} \frac{1}{n} \left( 1 + \frac{2}{1 + \sqrt{2}} + \frac{3}{1 + \sqrt{2} + \sqrt{3}} + \dots + \frac{n}{1 + \sqrt{2} + \sqrt{3} + \dots + \sqrt{n}} \right)
\]](http://latex.artofproblemsolving.com/e/d/7/ed7d17c7a2ff9b059c61331ce852b719f9f6826d.png)
![\[
\int_{0}^{2022\pi}\sqrt{\frac{\cos{2x}-\cos{2\pi}(2022)}{\cos{2x}+1}}dx
\]](http://latex.artofproblemsolving.com/4/1/6/416b980d55c9cd9978061d451490bf70ca580c59.png)
