AoPS Academy
![$\{x_n\} \subseteq [0,1]$](http://latex.artofproblemsolving.com/8/e/4/8e4206325387485d73765c9d7070ecb1ce18ff60.png)
, there exists some constant 
, for any 
, among the sequence 
and 
could be chosen to satisfy 
and 
.
colors such that no two adjacent vertices share the same color?![\[
\text{P} \subseteq \text{NP}.
\]](http://latex.artofproblemsolving.com/0/4/d/04d0eb775f943e138ec10566878d487e8a42f04e.png)
NP?
is NP-complete if:
(it is in NP, meaning solutions can be verified in polynomial time). in polynomial time (this means if you can solve efficiently, you can solve all NP problems efficiently).![\[
\zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s}
\]](http://latex.artofproblemsolving.com/7/e/d/7ed86c4c6a62ac6eb92a8ae9a809dda200bccdd0.png)
for complex numbers 
with a real part greater than 1. This infinite series converges for 
, but the Riemann zeta function is also analytically continued to other values of , except for 
, where it has a pole. where 
, lie on the "critical line," where the real part of is 
. In other words, all such zeros should be of the form 
, where 
is a real number.
and applies the following steps: is even, divide it by 2. is odd, multiply it by 3 and add 1.
:![\[
6 \rightarrow 3 \rightarrow 10 \rightarrow 5 \rightarrow 16 \rightarrow 8 \rightarrow 4 \rightarrow 2 \rightarrow 1
\]](http://latex.artofproblemsolving.com/6/1/b/61b4e2842630cb74b4a2c25ac9cd1b23216d1ca0.png)
While this looks simple, the Collatz Conjecture remains unsolved. The sequence has been verified for a large range of numbers, but a general proof that every positive integer eventually reaches 1 is still elusive. No one has been able to prove that the sequence always terminates, nor has anyone been able to find a counterexample. The conjecture is puzzling because it involves very basic operations but exhibits unpredictable behavior.![\[
\text{Every even integer greater than 2 is the sum of two prime numbers.}
\]](http://latex.artofproblemsolving.com/4/9/0/4908b423ef1d61a00fb8fdc5e2b90fbdc426ff54.png)
, there will always be some twin prime pair larger than .
be a positive semi-definite matrix. Prove that the matrix 
, is also positive semi-definite for all 
.
arcsin
is semi-definite. In fact it's not hard to show that it actually works for any analytic function 
with positive Maclaurin coefficients.
![$C=[f(x^{i+j})]_{i,j}$](http://latex.artofproblemsolving.com/9/c/3/9c3cf1fef646f73beb8f124872f9b11f60346843.png)
?
?
denotes the matrix, then for 
, where 
we have 
. So we only need to show that 
is semidefinite positive, for 
. But this is just a trivial computation.
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Bracket 3![$$\int_0^{1}\frac{dx}{1-\sqrt[3]{1-\sqrt{x}}}$$](http://latex.artofproblemsolving.com/d/5/e/d5e8db00985944b2eca341e37a94e34661cb5c86.png)
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![$$\int_0^{2\pi}\sqrt[4]{\cos^4(x)-\cos(2x)}dx$$](http://latex.artofproblemsolving.com/7/e/e/7eebb4a488fec702e0130a253ef87f0f6dfedb2d.png)
2. ![$$\int_0^1 \frac{x^5}{\sqrt[3]{1-x^3}}dx$$](http://latex.artofproblemsolving.com/6/0/5/605bb9d81d0ccff54a276c9f7bf1c1726c136398.png)
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2. ![$$\int_0^1 \frac{dx}{(\sqrt{x}+\sqrt[3]{x})^2}$$](http://latex.artofproblemsolving.com/6/9/8/69869c0ed754c0dfacf412ac6411742ffa164a6d.png)
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