[/quote]
,
be a positive integer. Prove that there exist three permutations 
,
,
of 
such that ![\[\left|\sqrt{a_k}+\sqrt{b_k}+\sqrt{c_k}-2\sqrt{N}\right|<2023\]](http://latex.artofproblemsolving.com/9/8/0/98097fb721722b9103ef65d72322956877e262cb.png)
for every 
.
.
,
has at most 
be the sum of the digits of 
such that there are infinite positive integers 
.
be three diameters of the circumcircle of an acute triangle 
.
be an arbitrary point in the interior of 
,
be the orthogonal projection of 
,
be the point such that 
is the midpoint of 
,
be the point such that 
is the midpoint of 
,
be the point such that 
is the midpoint of 
.
is similar to triangle 
such that if 
and 
are relatively prime divisors of 
,
divides 
is an infinite sequence of integers satisfying the following two conditions:
divides 
for 
such that 
for all 
such that 
for all 
with complex coefficients for which all complex roots of the polynomials 
have absolute value 1.
Show that there exists a constant 
such that for all positive integer 
![\[\sum_{k\le n^{\varepsilon}}(n\text{ mod } k)>cn^{2\varepsilon}.\]](http://latex.artofproblemsolving.com/f/e/9/fe933e886e28c2ecf1fa880d0e650ac7a2998aaa.png)
Proposed by Cheng Jiang
such that 
be an isosceles trapezoid with 
.
.
and 
the circumcircles of the triangles 
and 
,
with the tangent to 
is tangent to 
Y by LostInBali, Pengu14, aidan0626, Alex-131, Aaron_Q, pi-ay
I'm a rising ninth grader who wasn't in the school math league this year, and basically put aside comp math for a year. Unfortunately, that means that now that I'm in high school and having the epiphany about how important comp math actually is, and how much it would help my chances of getting involved in other math-related programs. In addition, I do enjoy math in general, and suspect that things like the AMCs are probably going to be some of the best practice I can get. What this all means is that I'm trying to go from mediocre to orz, 2 years after I probably should have started if I wanted to be any good.
So my question is: how do I get good at comp math?
This year, my scores on AMC 10 (and these are the highest I've ever gotten) were a 73.5 and an 82.5 (AMC 8 was 21/25, but that doesn't matter much). This is not good enough to qualify for AIME, and I probably need to raise my performance on each by at least 10 points. I've been decently good in the past at Number Theory, but I need to work on Geo and Combinatorics, and I'm trying to find the best resources to do that. My biggest flaw is probably not knowing many algorithms like Stars and Bars, and the path is clear here (learn them) but I'm still not sure which ones I need to know.
I'm aware that some of this advice is going to be something like "Practice 5 hours a day and start hardgrinding" or something along those lines. Unfortunately, I have other extracurriculars I need to balance, and for me, time is a limiting resource. My parents are somewhat frowning upon me doing a lot of comp math, which limits my time as well. I have neither the time nor motivation to do more than an hour a day, and in practice, I don't think I can be doing that consistently. As such, I would need to make that time count.
I know this is a very general question, and that aops is chock-full of detailed advice for math competitions. However, I'd appreciate it if anyone here could help me out, or show me the best resources I should use to get started. What mocks are any good, or what textbooks should I use? Where do I get the best practice with the shortest time? Is there some place I can find a list of useful formulas that have appeared in math comps before?
All advice is welcome!
So my question is: how do I get good at comp math?
This year, my scores on AMC 10 (and these are the highest I've ever gotten) were a 73.5 and an 82.5 (AMC 8 was 21/25, but that doesn't matter much). This is not good enough to qualify for AIME, and I probably need to raise my performance on each by at least 10 points. I've been decently good in the past at Number Theory, but I need to work on Geo and Combinatorics, and I'm trying to find the best resources to do that. My biggest flaw is probably not knowing many algorithms like Stars and Bars, and the path is clear here (learn them) but I'm still not sure which ones I need to know.
I'm aware that some of this advice is going to be something like "Practice 5 hours a day and start hardgrinding" or something along those lines. Unfortunately, I have other extracurriculars I need to balance, and for me, time is a limiting resource. My parents are somewhat frowning upon me doing a lot of comp math, which limits my time as well. I have neither the time nor motivation to do more than an hour a day, and in practice, I don't think I can be doing that consistently. As such, I would need to make that time count.
I know this is a very general question, and that aops is chock-full of detailed advice for math competitions. However, I'd appreciate it if anyone here could help me out, or show me the best resources I should use to get started. What mocks are any good, or what textbooks should I use? Where do I get the best practice with the shortest time? Is there some place I can find a list of useful formulas that have appeared in math comps before?
All advice is welcome!
