[/quote]
,
be positive real numbers such that 
.
.
and 
Prove that
be a scalene triangle with circumcenter 
,
,
.
,
,
at points 
,
,
respectively. Let 
be the projection of 
.
intersects the circumcircle of 
.
and 
intersect 
and 
respectively. Prove that the lines 
,
,
are concurrent.
be positive real numbers such that : 
. Prove that : 
and 
such that 
.
,
,
satisfying 
.
be an acute triangle with 
.
be the centroid of triangle 
and let 
to side 
.
intersects the circumcircle 
of triangle 
.
be the point where 
intersects the circle 
,
and 
and circumcenter 
and 
is a parallelogram. Prove that 
.
be the set of positive real numbers. Determine all functions 
such that ![\[x \big(f(x) + f(y)\big) \geqslant \big(f(f(x)) + y\big) f(y)\]](http://latex.artofproblemsolving.com/e/7/1/e71b6e0d0b858eb46b81149f1e6be8c41e13d301.png)
for every 
.
![\[
(a + b + c) \left( \frac{a}{(b - c)^2} + \frac{b}{(c - a)^2} + \frac{c}{(a - b)^2} \right) > 4.
\]](http://latex.artofproblemsolving.com/2/0/9/209f9807481e7a1e97f023bf65a603cb462c62cb.png)
(Karl Czakler)
that can be written in the form![\[
n = \frac{a^2 - b^2}{b},
\]](http://latex.artofproblemsolving.com/f/9/0/f90a3ce2e1b51d6e1cb53a15cad6f36c2b30492a.png)
where 
Y by PikaPika999, KevinYang2.71, Alex-131, Awesomeness_in_a_bun, megarnie
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?
