AoPS Academy
be the orthocenter of acute triangle 
, let 
be the foot of the altitude from 
to 
, and let 
be the reflection of across 
. Suppose that the circumcircle of triangle 
intersects line at two distinct points 
and 
. Prove that is the midpoint of 
.
be a quadrilateral inscribed in circle 
with 
. Let 
and be the reflections of 
over lines 
and , respectively, and let be the intersection of lines 
and 
. Suppose that the circumcircle of 
meets at and 
, and the circumcircle of 
meets at and 
. Show 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 
such that among any 
positive real numbers 
with 
, there exist three that are the side lengths of an acute triangle.
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 
.
All I can do now is hope and pray for HCSSiM. So far I'm 0 for 3 for summer programs lol
All I can do now is hope and pray for HCSSiM. So far I'm 0 for 3 for summer programs lol
All I can do now is hope and pray for HCSSiM. So far I'm 0 for 3 for summer programs lol
All I can do now is hope and pray for HCSSiM. So far I'm 0 for 3 for summer programs lol
All I can do now is hope and pray for HCSSiM. So far I'm 0 for 3 for summer programs lol
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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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Bracket 2
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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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