[/quote]
,
such that 
coprime, 
and 
.
,
and 
be the circumcenter and orthocenter, respectively. Let 
and 
be the midpoints of 
and 
,
and 
be the feet of the perpendiculars from 
and 
to the opposite sides, respectively. Show that if 
is the intersection of 
and 
,
is perpendicular to 
.
.
.![$(a+b+c)^4(ab+bc+ca) - 9\sum{\frac{a}{c}} \ge 54[(a+b)(b+c)(c+a)+abc-1]$](http://latex.artofproblemsolving.com/d/3/c/d3cf439c79cb84ca23083e51b6190ab5118f34e8.png)
denote the sum of all integers 
such that 
and exactly one of 
and 
is divisible by 
.
.
.
.
and 
,
of positive integers that satisfy
.
![\[\sqrt x - \frac 1y - 2w + 3z = 1,\]](http://latex.artofproblemsolving.com/d/0/f/d0ff4819db851e6c12f738dcdb6b81bbd2ed0ca7.png)
![\[x + \frac{1}{y^2} - 4w^2 - 9z^2 = 3,\]](http://latex.artofproblemsolving.com/6/5/3/653c1547620f85ff761817a5cc333f16bce9c4c6.png)
![\[x \sqrt x - \frac{1}{y^3} - 8w^3 + 27z^3 = -5,\]](http://latex.artofproblemsolving.com/1/b/9/1b9c2ed383acc931aa3607ba82031e182e18f3d1.png)
![\[x^2 + \frac{1}{y^4} - 16w^4 - 81z^4 = 15.\]](http://latex.artofproblemsolving.com/e/0/3/e032ce109cd74bbae47dc6bca04dcab2d8f9ef29.png)
and 
.
.
and 
intersect at point 
is the point on segment 
such that 
.
is isosceles.
![\[\sqrt{a+h_B}+\sqrt{b+h_C}+\sqrt{c+h_A}=\sqrt{a+h_C}+\sqrt{b+h_A}+\sqrt{c+h_B}\]](http://latex.artofproblemsolving.com/5/d/6/5d6881be29327bcdc3e7bd76c221968578c7f50b.png)
is isosceles.
,
,
are the roots of 
,
.
distinct primes?
Y by Adventure10
Hi guys,
I'm in 8th grade, and I plan to take the AMC 10/12 next year, and the AIME(if I make it). I want to know what is the best preparation for these two tests.
I know Aops has all the tests from the AMC series, and the AIME's, and I have done some problems from those books, and they don't seem all to hard. I have Aops Intermediate textbooks(except calculus), and I want to know if they will become a good preparation for AMC\AIME.I would also like to know the level you would have to be at if you want to be if you want to score well on these tests( I scored in the top 30 in my state for MATHCOUNTS, and our school was the top scoring public school in the state( yay, 6th place!)
If you have any comments or suggestions, please reply!!
Thank you guys!!
I'm in 8th grade, and I plan to take the AMC 10/12 next year, and the AIME(if I make it). I want to know what is the best preparation for these two tests.
I know Aops has all the tests from the AMC series, and the AIME's, and I have done some problems from those books, and they don't seem all to hard. I have Aops Intermediate textbooks(except calculus), and I want to know if they will become a good preparation for AMC\AIME.I would also like to know the level you would have to be at if you want to be if you want to score well on these tests( I scored in the top 30 in my state for MATHCOUNTS, and our school was the top scoring public school in the state( yay, 6th place!)
If you have any comments or suggestions, please reply!!
Thank you guys!!
