[/quote]
,
be an acute, non-isosceles triangle, 
be its heights and 
its circumcircle. 
cuts the circumcircle at points 
,
being between points 
.
be the midpoints of the major and the minor arc 
,
cuts 
.
and 
cut again 
at points 
and 
,
is the line passing through the circumcenters of triangles 
and 
the parallel lines through 
to 
meets 
at 
and 
meets 
at 
,
are concyclic.
such that: 
.![$$ \sqrt[3]{4} \geq a^2-ab+b^2 \geq 1$$](http://latex.artofproblemsolving.com/e/a/2/ea2874bcb691804ea1cc026a5cba87f04d17f64e.png)
be real numbers less than 2.Then prove that 
such that 
for all 
is called 
if there exists a permutation 
of the numbers 
,
and 
have different parities for every 
;
the sum 
is a quadratic residue modulo 
for every 
.
and 
Prove that
,
.
lies on segment 
such that 
.
and 
be the circumcenter and incenter of triangle 
,
and 
be the circumcenter and incenter of triangle 
,
and 
intersect at point 
.
.
be positive reals. Prove that there is a cyclic permutation 
of 
such that the inequality:
holds for all positive real numbers 
and 
.
,
is acute, 
,
.
be the intersection of the tangent to the circumcircle of 
and the perpendicular bisector of segment 
.
,
.
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 
.
,
.
