such that
Prove that
Let 
Prove that![$$ (a+b)(a+1)(b+1) \leq \frac{3}{2}+\sqrt[3]{6}+\sqrt[3]{36}$$](http://latex.artofproblemsolving.com/f/5/e/f5e68204507aa9c3e71c28844bbee3a804546b94.png)
be a positive integer larger than 
.
of subsets of 
having the following properties:
distinct sets of 
and 
be positive integers such that 
divides 
.
.
![$f\in C^1[a,b]$](http://latex.artofproblemsolving.com/8/4/8/848da12534d564560fe1cb016142159234be98b4.png)
,
and suppose that 
,
,
for all ![$x\in [a,b]$](http://latex.artofproblemsolving.com/7/8/d/78d9dcd969a2c45f7cc9234aa1377b6ec1c2d5d3.png)
.
for all 
![$f \in C([0,1];[0,1])$](http://latex.artofproblemsolving.com/4/7/5/475518b7db0b787dd90b93f602c7b912c663fba3.png)
bijective, 
and ![$(a_k) \in [0,1]^\mathbb N$](http://latex.artofproblemsolving.com/a/d/8/ad8a80d847c60ee8e916a1f9b08b7b03ec17c814.png)
with 
converge.
converge?
is called perfect if the sum of its positive divisors 
is twice 
.
is a perfect number since the sum of its positive divisors is 
,![\[
\sum_{p|n} \frac{1}{p + 1} < \ln 2 < \sum_{p|n} \frac{1}{p - 1}
\]](http://latex.artofproblemsolving.com/0/c/0/0c08f527e87f7aae4ca5226f19bd084249551fe3.png)
where the sums are taken over all prime divisors 
of 
![$f\in C[0,b]$](http://latex.artofproblemsolving.com/8/9/8/898aec0300888e562794b7a551be34f5c2676236.png)
,
and let 
be periodic with period 
has a limit as 
and
be an even positive integer. Let 
be a monic, real polynomial of degree 
;
for some real coefficients 
.
for all integers 
.
for which 
.
where 
,
,
are nonnegative integers.
,
,
and 
for 
.
and give an example where 
.
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)
be the set of all 
matrices with at most two entries in each row equal to 
.
.
denotes the determinant of the matrix 
Y by
Let 
be polygons, no two of which are similar. Show that there are polygons 
where 
is similar to 
so that for no 
does contain a polygon that's congruent to 
.
Note. Here, the word "contain" means for the construction we have, we cannot select a size for so that is wholly contained in , and so it does not intersect the edges of at all.
be polygons, no two of which are similar. Show that there are polygons 
where 
is similar to 
so that for no 
does contain a polygon that's congruent to 
.Note. Here, the word "contain" means for the construction we have, we cannot select a size for
so that is wholly contained in , and so it does not intersect the edges of at all.This post has been edited 1 time. Last edited by somebodyyouusedtoknow, Apr 26, 2025, 11:41 PM
Reason: Added clarity for the notes
Reason: Added clarity for the notes
is obvious; for fixed 
consider the largest copy of 
inside it and slightly enlarge it to obtain 
.
.
with 
for all 
we seek for appropriate 
.![$i\in [n]:= \{1, \ldots, n\}$](http://latex.artofproblemsolving.com/5/5/7/5572574af6c66362b508aebea9989529caa5811f.png)
define 
as the largest copy of 
inside 
as the smallest copy of ![$i\in [n]$](http://latex.artofproblemsolving.com/e/e/5/ee51c5c5eb09111cc774a00e9970074ad9f9058c.png)
.![$\max_{i\in [n]} R_i^- \subsetneq \min_{i\in [n]}R_i^+$](http://latex.artofproblemsolving.com/5/0/1/501e5e0ead06031f5c49050fed96583c031780da.png)
.
.
