[/quote]
,
be a triangle with circumcircle 
and 
a line without common points with 
the foot of the perpendicular from the center of 
intersect 
different from 
,
and 
have a common point different from 
have the property that both![\[
\left\lfloor \sqrt{2n - 1} \right\rfloor \quad \text{and} \quad \left\lfloor \sqrt{3n + 2} \right\rfloor
\]](http://latex.artofproblemsolving.com/d/e/c/dec527c4673470e7023d6aed00fac90268bd2fcc.png)
are consecutive perfect squares.
such that
is an integer.
,
be the foot of the altitude from 
.
be a point in the interior of the segment 
.
be the point on the segment 
such that 
.
be the point on the segment 
such that 
.
be the point of intersection of 
and 
.
.
be a point on segment 
.
be a fixed circle passing through 
be a variable point on 
be the intersection of the tangent to the circumcircle of 
at 
and the tangent to the circumcircle of 
at 
.
be the set of positive integers. A function 
satisfies the equation ![\[\underbrace{f(f(\ldots f}_{f(n)\text{ times}}(n)\ldots))=\frac{n^2}{f(f(n))}\]](http://latex.artofproblemsolving.com/2/e/7/2e73ff2f75b103e9d6a4004a42e4ed7f4ee64a67.png)
for all positive integers 
.
,
,
,
.
such that for every 
![\[
f_n(x) = \frac{x}{1 + nx}, \quad \text{if and only if } f(x) = \frac{x}{1 + x}, \quad \forall x \geq 0.
\]](http://latex.artofproblemsolving.com/0/f/c/0fc36d9264eb7e103128c489aeae521a859c1fd4.png)
(For 
represents 
.
be the sum of the remainders when 
.
.
.
such that 
and 
have the same number of 
'
.
,
,
,
(in this order, from left to right). Determine whether the set of all beautiful integers is finite.
Y by PikaPika999, KevinYang2.71, Alex-131, Awesomeness_in_a_bun, megarnie, bjump
I see a lot of people on aops who get "lucky" sometimes and get "unlucky" sometimes, and i think i have had lucky and unlucky times before.
but i want to know what exactly causes you get lucky or unlucky? i heard people saying the more u work the luckier u get. but i bombed amc and aime this year despite working alot (i mean mainly olympiad problems but i did do mocks and computational problems before
). so why do u just get lucky or unlucky. Is it possible to get lucky if ur under pressure too much, like if u care too much about ur results?
Also, on another topic this year my aime testing room (i took it at my school) was super loud and kids kept walking through and talking. it's like the only free room in the school, the other one smells really bad and has 0 ventilation and gives lung cancer. i get distracted sometimes when people talk so loudly and for some reason maybe pressure i ended up doing alot worse than my usual mocks and missed usamo :/. if i solved 1 more problem wouldve made. sucks cuz i mainly grinded olympiads. so what do u guys think i should do, should i take it at the aops academy near me? or should i just stay at my school?
has this situation happened for anyone else, what did u do about it?
but i want to know what exactly causes you get lucky or unlucky? i heard people saying the more u work the luckier u get. but i bombed amc and aime this year despite working alot (i mean mainly olympiad problems but i did do mocks and computational problems before
). so why do u just get lucky or unlucky. Is it possible to get lucky if ur under pressure too much, like if u care too much about ur results?Also, on another topic this year my aime testing room (i took it at my school) was super loud and kids kept walking through and talking. it's like the only free room in the school, the other one smells really bad and has 0 ventilation and gives lung cancer. i get distracted sometimes when people talk so loudly and for some reason maybe pressure i ended up doing alot worse than my usual mocks and missed usamo :/. if i solved 1 more problem wouldve made. sucks cuz i mainly grinded olympiads. so what do u guys think i should do, should i take it at the aops academy near me? or should i just stay at my school?
has this situation happened for anyone else, what did u do about it?
