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and let 
and 
be permutations of 
. Suppose there exists a permutation 
such that 
for all 
in .
is the number of possible pairs of permutations 
, find the remainder when is divided by 1000.
and 
Prove that 
be real numbers such that 
. Find the minimum of 
and 
Prove that 
positive integer arrays 
, and they are all infinity arrays.
, that for every integer 
![\[
\sqrt[3]{\sqrt[3]{x - \frac{3}{8}} - \frac{3}{8}} = x^3 + \frac{3}{8}
\]](http://latex.artofproblemsolving.com/b/b/9/bb99cd588b94b5716738e6e9ea01d15377b6efef.png)
has exactly two real positive solutions 
and 
. Compute 
.
, the external bisectors of 
and 
meet at a point 
. Prove that the circumcentre of 
and the points 
lie on the same straight line.
. How many complex numbers 
are there such that 
and 
is an integer. be reals such that 
. Prove that
Let be reals such that 
. Prove that
and 
for each positive integer 
, what is the value of 
?
be a positive integer such that 
, and let the set![\[
A = \{a_1, a_2, a_3, \ldots, a_m\}
\]](http://latex.artofproblemsolving.com/1/7/9/179687a134d5495ffff0b44fe763a2bb8b3ddd9c.png)
consist of distinct positive integers not exceeding 2025. Suppose that for every 
, with 
, if 
, then 
as well. Prove that![\[
\frac{a_1 + a_2 + a_3 + \cdots + a_m}{m} \geq 1013.
\]](http://latex.artofproblemsolving.com/c/0/d/c0d190f78d94deac1b947ba57a2de51d0b06f0bd.png)
for which I can find nonnegative integers 
such that for all indices 
, 
divides 
?
...") for which I there exists nonnegative integers such that for all indices , divides ?
integers - take the set of all integers with a digit sum of 
in base , then append 
to the list.
. Points 
, 
and 
are marked on the sides 
, 
, 
, respectively. It turned out that 
is perpendicular to , 
is perpendicular to and 
. Prove that 
is perpendicular to .
and 
Prove that
be real numbers such that 
Prove that
and ![$e\in [-2,2]$](http://latex.artofproblemsolving.com/b/b/7/bb78c9dedb18af5e6a904f5492917921e41a7ebc.png)
such that 
Find the value of 
,
and 
.