[/quote]
,
be the sequence of all natural numbers which are sum of squares of two natural numbers.
which 
.
intersect at 
,
tangent to the extensions of 
at 
respectively. Circle 
passes through points 
,
.
.
such that 
for reals 
.
but "teS", "sorat", "setarcoS" 
.
and 
as follows: take the string with the first term that appears first in "Socrates" (if both have the same first term, take the shorter one) and WLOG let it be 
"Socrate" and "cra" + "ates" 
such that
.
is still the same.
that can be written in the form
where 
are pairwise coprime positive integers such that 
,
,
.
and 
: Prove that :
such that 
.
and 
Prove that
be an odd integer greater than 1 and let 
be integers. For each permutation 
of 
,
.
of 
is a divisor of 
.
and 
be two infinite sequences of integers. Prove that there exists an infinite sequence of integers 
such that for any positive integer 
,![\[
\sum_{k|n} k \cdot z_k^{\frac{n}{k}} = \left( \sum_{k|n} k \cdot x_k^{\frac{n}{k}} \right) \cdot \left( \sum_{k|n} k \cdot y_k^{\frac{n}{k}} \right).
\]](http://latex.artofproblemsolving.com/2/5/9/259ee32c89befe7a3ee2797e0d5b1df1c3066cf6.png)
,
,
is the Euler's totient theorem, which calculates the number of integers less than 
for 
.
be the remainder 
leaves when you divide it by 
.![$\lfloor \frac{x + \sqrt[r]{r^2} + r \ln(r)}{r} \rfloor$](http://latex.artofproblemsolving.com/c/d/3/cd3f7d917b0ec944ec3017e1f737c538e1edd10e.png)
.
unless of your age can be written as 
.
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?
