[/quote]
,
be a triangle with 
.
and 
lie on sides 
and 
,
and 
.
and 
meet at 
.
,
,
,
to all have integer lengths.
such that the equality 
holds for all pairs of real numbers 
.
is an obvious solution. Now, let 
.
or 
.
.
,
:
,
be an integer. Find all sequences 
satisfying the following conditions:![\[ \text{ (a) } a_i \in \left\{0,1\right\} \text{ for all } 1 \leq i \leq n^2 + n;
\]](http://latex.artofproblemsolving.com/3/c/5/3c509ec2e9013e8d3be492c8eb44a7c33841b74e.png)
![\[ \text{ (b) } a_{i + 1} + a_{i + 2} + \ldots + a_{i + n} < a_{i + n + 1} + a_{i + n + 2} + \ldots + a_{i + 2n} \text{ for all } 0 \leq i \leq n^2 - n.
\]](http://latex.artofproblemsolving.com/9/7/d/97d2a467d1c0dc8594ec024c3bb9b8c87ee85b19.png)
.
be the midpoint of 
be the midpoint of 
and let 
be the centroid of the triangle 
be a circle through 
.
and 
are collinear.
with integer coefficients such that ![$P\left(\sqrt[3]5+\sqrt[3]{25}\right)=5+\sqrt[3]5$](http://latex.artofproblemsolving.com/5/8/e/58ec2a921cd988bd5ae017a8653544ddd758225f.png)
.
-
.
,
in 
such that the sets ![\[ A_j=\{x+t_j\mid x\in A\},\qquad j=1,2,\ldots,100 \]](http://latex.artofproblemsolving.com/d/5/c/d5c2a7b943e8c7200361698f3eb6f83700ad68d6.png)
are pairwise disjoint.
such that for every index 
,
,
is a multiple of 
.
and 
be the points in which the median and the angle bisector, respectively, at 
.
and 
meets 
and 
,
the point in which the perpendicular at 
produced.
is perpendicular to 
Y by
im in 7th grade and took the AMC 10 A/B with absouletely nauseating score, which i will not reveal. I wasnt even close to AIME frankly. My goals are the following:
7th grade: AMC 8 - DHR
8th grade:AIME qual, AMC 8 Perfect
9th grade: AMC 10 DHR maybe?, AIME 7+
10th grade: USAJMO, AIME 12+, AMC 10 DHR
11th grade: USAMO, AIME 12+, AMC 12 DHR
12th grade: USAMO, AIME great score, AMC 12 perfect or close?
These are the goals that i want to achieve. I will do literally anything to achieve them. Can someone please give me a good tip so i can follow it for the next 5 years of my life to become a 3 time USAMO qual and a 5 time AIME qual, and have an perfect AMC 8 under my belt?
7th grade: AMC 8 - DHR
8th grade:AIME qual, AMC 8 Perfect
9th grade: AMC 10 DHR maybe?, AIME 7+
10th grade: USAJMO, AIME 12+, AMC 10 DHR
11th grade: USAMO, AIME 12+, AMC 12 DHR
12th grade: USAMO, AIME great score, AMC 12 perfect or close?
These are the goals that i want to achieve. I will do literally anything to achieve them. Can someone please give me a good tip so i can follow it for the next 5 years of my life to become a 3 time USAMO qual and a 5 time AIME qual, and have an perfect AMC 8 under my belt?
