[/quote]
,
and 
be positive integers with 
.
,
.
be the orthocenter of acute triangle 
,
be the foot of the altitude from 
to 
,
.
intersects line 
and 
.
.
be positive integers with 
.
be a polynomial of degree 
such that the polynomial 
divides 
is zero. Prove that 
be positive integers. Prove that there exists a positive integer 
such that for every odd integer 
,
are all greater than 
such that![\[
f(xy) = f(x)f(y) \;-\; f(x + y) \;+\; 1,
\quad \forall x, y \in \mathbb{R}.
\]](http://latex.artofproblemsolving.com/7/a/5/7a51a7be2306d18f4e80db1f5aac6e9bbf25c25c.png)
be a positive integer. For every positive integer 
the sequence 
is defined, where 
are integers. Among these 
be a sequence of real numbers with 
.
of a permutation 
of 
to be the minima of the sum 
over all real numbers 
.
,
,
be a nonempty set of positive integers. We say that a positive integer 
such that for any positive integer 
,
.
lie respectively on sides 
of a triangle 
.
and 
cross at 
.
and 
cross again at point 
.
satisfying 
and their sum is an even number, consider the quadratic polynomial:![\[
P(x) = x^2 - (m^2 - m + 1)x + (m^2 - n^2 - m)(n^2 + 1).
\]](http://latex.artofproblemsolving.com/d/5/1/d51aaddfddfe06cc91abfdd3476a155770a8a5fd.png)
are positive integers but are not perfect squares.
such that![\[
ad = bc, \quad a + d = 2^k, \quad b + c = 2^m
\]](http://latex.artofproblemsolving.com/1/1/d/11d16732e3a77af46ad17adecf104a442a5d7f67.png)
for some positive integers 
and 
.
and intouch triangle 
.
onto 
.
,
respectively. Finally, let the rays 
,
meet the circumcircle of 
in 
meet on the circumcircle of 
Y by Adventure10, Mango247
Even though there have been threads upon threads discussing preparation for the AMC, it's still not 100% clear to me what I should be doing.
Last year, I got a 114 with minimal preparation on the AMC 10 with two stupid mistakes that costed me an AIME invite. This year, my goal is to get in the Distinguished honor roll (about 132 at least for AMC A).
I've been doing more preparation this year than last year especially with Aops Volume I (I also have II but I haven't been using it often). Every day when I'm studying, I'll feel motivated for about 90-120 minutes, but then I'll hit a wall and won't want to study unless I force myself to.
So a few questions here...
1. Should I just study until I don't feel like learning more for the day?
2. How much would doing Volume II help on the AMC 10? Should I do it anyway if I think I'll make it to AIME?
3. Does doing past AMC problems help a lot? (even AMC 12 problems or AIME problems?)
Any other advice would be appreciated too.
Last year, I got a 114 with minimal preparation on the AMC 10 with two stupid mistakes that costed me an AIME invite. This year, my goal is to get in the Distinguished honor roll (about 132 at least for AMC A).
I've been doing more preparation this year than last year especially with Aops Volume I (I also have II but I haven't been using it often). Every day when I'm studying, I'll feel motivated for about 90-120 minutes, but then I'll hit a wall and won't want to study unless I force myself to.
So a few questions here...
1. Should I just study until I don't feel like learning more for the day?
2. How much would doing Volume II help on the AMC 10? Should I do it anyway if I think I'll make it to AIME?
3. Does doing past AMC problems help a lot? (even AMC 12 problems or AIME problems?)
Any other advice would be appreciated too.
)
