red marbles, 
yellow marbles, 
green marbles, and 
blue marbles. If Amanda takes the marbles one by one without replacing them until the 
marble. Then the probability that the 
marble is red is?
be a metric space and 
be a function such that 
.
Prove that for all 
,
exists, where 
is 
.
Prove that the amount of the limit does not depend on choosing 
.
![\[\iint_{R} \sin(xy) \,dx\,dy, \quad R = \left[0, \frac{\pi}{2}\right] \times \left[0, \frac{\pi}{2}\right]\]](http://latex.artofproblemsolving.com/a/2/0/a205868dd99f4794de53ad1db50f34b17c70a923.png)
![\[y''+by'+cy=g(t)\]](http://latex.artofproblemsolving.com/7/8/8/7882dc954ab6dba2445b68d370d630ce1c66a6db.png)
with 
,
are constants has the form ![\[y(t)=\int^{t}_{t_0}K(t-s)g(s)ds\,\]](http://latex.artofproblemsolving.com/6/3/c/63c34df1c0013bf659bc76506a93873a5f879070.png)
and use variation of parameters in each case to find the solution. Is there any short way to calculate solution?
, 
,
be a positive sequence and let 
be a constant such that ![\[ \sum_{k=1}^{n-1} \lambda^2_k < K \lambda^2_n \;(n=1,2,...).\]](http://latex.artofproblemsolving.com/5/3/c/53ccb27390a368e1a74afae1da4f7566ad86af8a.png)
Prove that there exists a constant 
such that ![\[ \sum_{k=1}^{n-1} \lambda_k < K' \lambda_n \;(n=1,2,...).\]](http://latex.artofproblemsolving.com/e/7/4/e744b1b1179f879dbea84a53622afc5ca8e45fe2.png)
be a real function such that![\[ \lim_{x \rightarrow +\infty} \frac{f(x)}{e^x}=1\]](http://latex.artofproblemsolving.com/7/f/e/7fe200614381665ef11aed463fbbee51f8461bb3.png)
for all sufficiently large 
.![\[ \lim_{x \rightarrow +\infty} \frac{f'(x)}{e^x}=1.\]](http://latex.artofproblemsolving.com/0/9/6/096b5b71111efdd7565d60424240e20ee3251382.png)
be real numbers. Find the ranges of 
converges.
is defined on the interval 
.
is said to be a fixed point of 
.
and 
.
,
.
be the set of continuous functions ![$f: [0,1] \to (0, \infty )$](http://latex.artofproblemsolving.com/4/5/0/45018aa878524f722851ddc968a4f6aa10310bd5.png)
such that 
.
,
Determine 
.
Y by PikaPika999, Sedro
Hi people!
I have been and aops user for only a few months and not an active one either, just a user checking other posts and competitions. This is actually my first post. I am in 7th grade and only this year I have actually started locking in on competition math, before that, I didn’t really care about Mathcounts and AMCs. So, I hone my problem-solving skills by lightly doing some math problems. I have signed up for a lot of math competitions this month and am struggling to prepare properly with the resources I have. Do you guys have any resources you use to prepare for math competitions and that are actually helpful to use to prepare for competitions in a short amount of time. I am looking for resources that first teach you stuff and give you problems, like guided notes and handouts. I have few aops books like “Competition Math for Middle School” and “Intermediate Algebra”, but they’re not enough. I have learnt Algebra 1 and started doing high school geometry and a bit of Algebra 2 at home. Do let me know if you have any suggestions!
I have been and aops user for only a few months and not an active one either, just a user checking other posts and competitions. This is actually my first post. I am in 7th grade and only this year I have actually started locking in on competition math, before that, I didn’t really care about Mathcounts and AMCs. So, I hone my problem-solving skills by lightly doing some math problems. I have signed up for a lot of math competitions this month and am struggling to prepare properly with the resources I have. Do you guys have any resources you use to prepare for math competitions and that are actually helpful to use to prepare for competitions in a short amount of time. I am looking for resources that first teach you stuff and give you problems, like guided notes and handouts. I have few aops books like “Competition Math for Middle School” and “Intermediate Algebra”, but they’re not enough. I have learnt Algebra 1 and started doing high school geometry and a bit of Algebra 2 at home. Do let me know if you have any suggestions!
