AoPS Academy
add up to zero. Find the maximum of 
in term of 
's, when 
's vary in real numbers such that 
.
be an odd prime and 
be integers so that the integers 
are divisible by . divides each of .
denote the number of positive divisors of the natural number 
. Find all the natural numbers such that 
.
Show that the following statements are equivalent:
there exist 
such that 
there exist 
such that 
be a positive sequence and let 
be a constant such that ![\[ \sum_{k=1}^{n-1} \lambda^2_k < K \lambda^2_n \;(n=1,2,...).\]](http://latex.artofproblemsolving.com/5/3/c/53ccb27390a368e1a74afae1da4f7566ad86af8a.png)
Prove that there exists a constant 
such that ![\[ \sum_{k=1}^{n-1} \lambda_k < K' \lambda_n \;(n=1,2,...).\]](http://latex.artofproblemsolving.com/e/7/4/e744b1b1179f879dbea84a53622afc5ca8e45fe2.png)
such that 
are two finite groups with 
then show that 
be real numbers in the open interval 
Show that there exist distinct indices 
such that 
are the side lengths of an acute triangle.
of class 
for which there exists an 
such that 
be a positive integer, let 
, and let 
be a complex root of 
. Prove that there exist polynomials 
and 
with integer coefficients such that 
.
2. 
3. 
Bracket 2
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3. 
Bracket 3![$$\int_0^{1}\frac{dx}{1-\sqrt[3]{1-\sqrt{x}}}$$](http://latex.artofproblemsolving.com/d/5/e/d5e8db00985944b2eca341e37a94e34661cb5c86.png)
2. 
3. 
Bracket 4
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3. 
Bracket 5
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Bracket 6
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Bracket 7
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Bracket 8
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3. 
![$$\int_0^{2\pi}\sqrt[4]{\cos^4(x)-\cos(2x)}dx$$](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)
3. 
Bracket 2
2. 
3. 
Bracket 3
2. ![$$\int_0^1 \frac{dx}{(\sqrt{x}+\sqrt[3]{x})^2}$$](http://latex.artofproblemsolving.com/6/9/8/69869c0ed754c0dfacf412ac6411742ffa164a6d.png)
3. 
Bracket 4
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Bracket 2
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5. 
