[/quote]
,
be a right triangle with a right angle in 
and a circumcenter 
.
and 
,
and 
lie in such a way that 
.
and 
be the projection of 
,
is half as long as 
be an isosceles triangle with 
and circumcircle 
.
perpendicular to 
.
be any point on 
with center 
intersects 
and the circumcircle 
.
and 
,
,
,
.
such that for every 
![\[
f_n(x) = \frac{x}{1 + nx}, \quad \text{if and only if } f(x) = \frac{x}{1 + x}, \quad \forall x \geq 0.
\]](http://latex.artofproblemsolving.com/0/f/c/0fc36d9264eb7e103128c489aeae521a859c1fd4.png)
(For 
represents 
.
is a tetrahedron whose triangular faces are congruent to one another. What is the least total surface area of a disphenoid whose faces are scalene triangles with integer side lengths?
be the incentre of a non-equilateral triangle 
be the 
-
be the reflection of 
be the reflection of line 
in 
.
,
and line 
analogously. Let 
where 
is the circumcentre of triangle 
and 
.
.
such that 
is a perfect square and 
is a prime number.
,
,
,![\[ \frac{a^3}{bc} + \frac{b^3}{ca} + \frac{c^3}{ab} \geq a+b+c \]](http://latex.artofproblemsolving.com/9/f/a/9fa6cca20b3620fbb32ee782196e96f6fffa0063.png)
such that for any real value of 
we have:
,
,![\[ f(x)f(yf(x))=f(x+y). \]](http://latex.artofproblemsolving.com/5/6/0/560049dd70944cafa0423e5f1229d215f78215a0.png)
ones are written on a blackboard. Grisha and Gleb are playing game, by taking turns; Grisha goes first. On one move Grisha squares some 
numbers on the board. On his move, Gleb picks a few (perhaps none) numbers on the board and increases each of them by 
.
moves on the board a number divisible by 
appears, Gleb wins, otherwise Grisha wins. Which of the players has a winning strategy?
be a nonempty subset of the points in the Cartesian plane such that for each 
exactly one of 
or 
also belongs to 
there is a line in the plane (possibly different lines for different 
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 
.
,
.
