[/quote]
,
such that 
be a fixed integer. The cells of an 
table are filled with the integers from 
to 
with each number appearing exactly once. Let 
be the number of unordered quadruples of cells on this board which form an axis-aligned rectangle, with the two smaller integers being on opposite vertices of this rectangle. Find the largest possible value of 
of 
touch 
at 
,
,
denote the line tangent to 
.
such that 
.
of 
.
and 
.![\[\sum_{i=1}^n (a_i-a_{i+1})^3\leq \frac{1}{2\sqrt{3}} (\sum_{i=1}^n a_i )^3,\]](http://latex.artofproblemsolving.com/8/1/0/810f07e3394370ce06aec1e7025bcf5cc5433d60.png)
where 
.
and walks randomly. The probability of any point reaching all adjacent points is equal. Find the probability of the ant reaching 
with the properties that 
and 
is a real number. Find the remainder when 
.
of 
,
be the sum of the elements of 
defined to be 
.
,
is 
,
and 
are relatively prime positive integers. Find 
with 
prime and ![\[a^p+b^p=p!.\]](http://latex.artofproblemsolving.com/f/e/b/feb27894b1f6497559eeb094cfe92256bc26688e.png)
be the incenter of triangle 
.
,
,
be the orthocenters of triangles 
,
,
,
,
,
,
and 
have a common tangent that meets 
and 
.
and 
than 
meets circle 
.
bisects angle 
.
Y by Awesome_Twin1, Pengu14, Sedro
Hi everyone,
I recently created a handout on modular arithmetic for a local math club. I thought it would help quite a lot with understanding basic properties, as modular arithmetic is a very popular intermediate step in number theory problems, so I decided to leave it here as a resource for anyone who needs it. Feel free to share it around, and hope it helps!
Sincerely,
MathCosine
I recently created a handout on modular arithmetic for a local math club. I thought it would help quite a lot with understanding basic properties, as modular arithmetic is a very popular intermediate step in number theory problems, so I decided to leave it here as a resource for anyone who needs it. Feel free to share it around, and hope it helps!
Sincerely,
MathCosine
if there exists 
such that 
.
,
.
