Prove that
[/quote]
such that 
.
Where 
![$ a,b \in [0 ,1] . $](http://latex.artofproblemsolving.com/9/1/9/91980d5ca5d97fe4b61797e6fbb378211674d9cd.png)
Prove that
and 
Prove that 
be real numbers such that 
.
Prove that 
denote the set of positive integers. Find all functions 
such that for positive integers 
and 
![\[f(a^2 + b^2) = f(a)f(b) \text{ and } f(a^2) = f(a)^2.\]](http://latex.artofproblemsolving.com/c/1/f/c1f8ffe04cfcc45b498f3931a4796d5c56dc04d0.png)
that satisfy![\[
f(x^2-y)+2yf(x)=f(f(x))+f(y)
\]](http://latex.artofproblemsolving.com/f/b/b/fbb693770cd90d8a144026608074122e54da5802.png)
for all 
.
and ![$e\in [-2,2]$](http://latex.artofproblemsolving.com/b/b/7/bb78c9dedb18af5e6a904f5492917921e41a7ebc.png)
such that 
Find the value of 
:
be the altitude and 
be the intersection point of the altitudes of triangle 
.
is symmetric to 
.
intersects 
and 
.
.
where 
and 
where 
.
lie on the same circle. If 
and 
,
,
or 11. However, I didn't know how to find the radius. Can anyone give me a hint?
Y by KevinYang2.71
I registered as an individual as an early bird. I'm sure some of you guys who are taking this exam too, have received an email containing logistics. In that, there is the Google doc "Your Guide to SMT 2025 Online". I have a question. There is the whole section about waivers, which I have to fill out before competing. According to the doc: "Students will not be permitted to compete if they do not have a completed waiver by the start of the competition.". My question is what are these waivers for, as I have not applied for financial aid. My parents hasn't received any email about the waivers. "Please allow a day for waivers to be sent to parents and for waiver completion to be reflected on the student dashboard." I also don't have any waiver completion notification on my comp.mt(student dashboard).
Please respond fast.
Please respond fast.
This post has been edited 6 times. Last edited by techb, Apr 8, 2025, 8:32 AM
