[/quote]
,
denote the set of positive integers.
such that for all 
,![\[f(mn)^2 = f(m^2)f(f(n))f(mf(n))\]](http://latex.artofproblemsolving.com/5/b/c/5bcd0f162c76b4177801933623724e576476fa88.png)
if and only if 
and 
.
Let 
and 
.
Let 
and 
be an odd natural number. We consider an 
grid which is made up of 
unit squares and 
edges. We colour each of these edges either 
or 
.
where 
.
is the circumcenter of triangle 
is the projection of point 
onto line 
.
,
are 
,
,
,
intersects 
and 
at points 
and 
,
.
be real numbers less than 2.Then prove that 
be an acute-angled triangle and let 
,
,
be the midpoints of 
,
,
respectively. Construct a circle, centered at the orthocenter of triangle 
to intersect the circle at 
,
to intersect the circle at 
and 
to intersect the circle at 
.
.
and 
be two sequences of integers such that 
and 
for all 
.
divides 
for an infinity of integers 
such that 
for all 
Prove that 
Where 
be positive real variables such that 
Prove that 
be chords in a circle of center 
with 
distinct , and with the lines 
and 
be the midpoints of 
and 
respectively . If 
, prove that 
.
Y by
Does the math many people on AOPS work for really make a large impact? I get that this is a place for math but I think I'm losing touch with my prior math skill. AMC and AIME are solving problems that have answers and just take practice but math as a job or just like an accountant doesn't need/needs a lot more than this type of math. I might switch to another stem category for college apps pathway. Also, many people who are already very good at competition math don't get good offers because there's people who are even better(note: i get the point of math isn't only college). I think if I work *hard* I'll be AIME qual but I think many people would have higher achievements with the same amount of work in another category. I think competition math really just trains for problem solving, unique thinking, and pattern recognition(aka IQ pts).
