by MrOrange, May 30, 2025, 2:11 PM 
on 
for which![\[
\lim_{R \to \infty} \int_{\|x\|_2 \leq R} f(x) \, dx = 0
\]](http://latex.artofproblemsolving.com/4/6/9/469f69ee93ae8d1de73c31ceb066253751aa9775.png)
and for which![\[
\lim_{R \to \infty} \int_{\|x\|_\infty \leq R} f(x) \, dx \text{ does not exist.}
\]](http://latex.artofproblemsolving.com/d/f/a/dfadbdd902375b9f67cd2dd0dbb494978bd9e75c.png)
by SomeonecoolLovesMaths, May 28, 2025, 11:27 AM 
be real numbers. Define 
and 
.
and hence find the limit.by Entrepreneur, May 27, 2025, 9:19 AM 
is differentiable and 
Show that for some 
![$f:[0,1]\to\mathbb R$](http://latex.artofproblemsolving.com/e/a/0/ea02b990406104357c11912a8bf57bea7eeb519d.png)
is differentiable with 
If ![$|f'(x)|\le f(x)\;\forall\;x\in[0,1],$](http://latex.artofproblemsolving.com/b/9/d/b9d6088657cd85b37190b98abac1d94a18bb5916.png)
then show that 
be the unit circle in the complex plane. Let 
be the map given by 
We define 
and 
for 
The smallest positive integer 
such that 
is called period of 
Determine the total number of points 
of period 
denote the set of natural numbers, and let 
be nine distinct tuples in 
Show that there are 
distinct elements in the set 
whose product is a perfect cube.
and let 
be positive integers such that 
Prove that 
and determine when equality holds.by hzbrl, May 27, 2025, 3:49 AM 
, where 
. Verify by direct substitution that 
satisfies the quadratic equation 
and deduce that the value of is 
.
. Show that 
, show that the value of 
is 
.by giginori, Dec 3, 2023, 12:28 AM be an even positive integer. Let 
be a monic, real polynomial of degree 
; that is to say, 
for some real coefficients 
. Suppose that 
for all integers 
such that 
. Find all other real numbers 
for which 
.by giginori, Dec 3, 2023, 12:28 AM , let 
. Find the smallest such that 
.by mathematics2004, Nov 2, 2021, 4:38 PM 
be the set of all 
matrices with at most two entries in each row equal to 
and all other entries equal to 
.
.
denotes the determinant of the matrix 
.by awesomemathlete, Dec 10, 2019, 1:19 AM 
where , 
, and 
are nonnegative integers.by j___d, Mar 6, 2017, 5:00 PM ![$f\in C[0,b]$](http://latex.artofproblemsolving.com/8/9/8/898aec0300888e562794b7a551be34f5c2676236.png)
, 
and let 
be periodic with period 
. Prove that 
has a limit as 
and
by j___d, Mar 6, 2017, 4:49 PM 
, 
, 
and 
for 
. Prove that 
and give an example where 
."Where wisdom and valor fail, all that remains is faith. . . And it can overcome all." -Toa Mata Tahu
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