[/quote]
,
be the unit circle in the complex plane. Let 
be the map given by 
.
and 
for 
.
such that 
is called the period of 
.
of period 
.
)
such that![\[
x+f(xy)=f(1+f(y))x
\]](http://latex.artofproblemsolving.com/4/3/3/433f95836dad1224754d480aff8432ef033a37cb.png)
for all 
.
be an acute scalene triangle with circumcircle 
.
and 
be interior points of the sides 
and 
,
is parallel to 
.
be a point on 
is parallel to 
and 
intersect at 
.
intersects 
.
.
of positive integers such that if 
,
divides 
.
a group of 
people is called a "beautifu
people know each other. Assume that every beautiful 
2 people such that if they leave the conference, there will be no beautiful 3-group remaining.
such that
for all 
.
positive real numbers satisfying 
,![$\sqrt[3]{\frac{a^3+b^3+c^3}{3}}+\frac{a+b+c}{9} \geq \frac{4}{3}$](http://latex.artofproblemsolving.com/c/b/3/cb39e7f838c99ed8a209c6844f3d2810838d1dfd.png)
.
denote the set of natural numbers, and let 
,
,
.
whose product is a perfect cube.
are written on the board.In each step,two random numbers 
and 
are chosen and deleted.Then,the number 
is written instead.What will be the number remained on the board after the last step.
Y by
Hello, I am going to be a 7th grader next year and I really want to qualify for USAJMO in 8th grade, so far I have these goals reached
1. AMC 10 Honor Roll A and B 2025
2. AMC 8 DHR and HR
3. AIME 3
This year on AIME something happened and I got a 3 on the AMC's I got a 105 on AMC 10 A and I got a 114 on AMC 10 B. I want to improve mostly on AIME but since the AMC 10 is coming up quicker what would you guys recommend for getting 110+ on both of the AMC 10's and getting a 6+ on AIME? So far I am only doing Alcumus and have no books so far.... Checking the table of contents on the books Alcumus provides the same topics. I was thinking to take WOOT 1 and AMC 10 Problem Series.
1. AMC 10 Honor Roll A and B 2025
2. AMC 8 DHR and HR
3. AIME 3
This year on AIME something happened and I got a 3
on the AMC's I got a 105 on AMC 10 A and I got a 114 on AMC 10 B. I want to improve mostly on AIME but since the AMC 10 is coming up quicker what would you guys recommend for getting 110+ on both of the AMC 10's and getting a 6+ on AIME? So far I am only doing Alcumus and have no books so far.... Checking the table of contents on the books Alcumus provides the same topics. I was thinking to take WOOT 1 and AMC 10 Problem Series.
