be an arbitrary triangle and 
is the circumcenter of 
.
lie on 
,
is tangent to circumcircle of 
.
is tangent to circumcircle of triangle 
.
with 
,
be an integer. In a configuration of an 
board, each of the 
cells contains an arrow, either pointing up, down, left, or right. Given a starting configuration, Turbo the snail starts in one of the cells of the board and travels from cell to cell. In each move, Turbo moves one square unit in the direction indicated by the arrow in her cell (possibly leaving the board). After each move, the arrows in all of the cells rotate 
counterclockwise. We call a cell good if, starting from that cell, Turbo visits each cell of the board exactly once, without leaving the board, and returns to her initial cell at the end. Determine, in terms of 
,
positive integers 
such that for every index 
,
,
is a multiple of 
.
and 
be positive integers satisfying the equation ![$(a, b) + [a, b]=2021^c$](http://latex.artofproblemsolving.com/2/4/6/246ccbb22acecb7adc72219afb8afb34c9b05101.png)
.
is a prime number, prove that the number 
is composite.
be the set of real numbers. Determine all functions 
such that ![\[ f(x^2 - y^2) = x f(x) - y f(y) \]](http://latex.artofproblemsolving.com/6/8/f/68f78e0df627f6ac04052433c01627bf16f0af8e.png)
for all pairs of real numbers 
and 
.
into 
or 
can sometimes be quite complex. That's why I’ve written a program to assist with this process.
be real numbers such that 
Prove that 
Where 
with odd integer side lengths is divided into small rectangles with integer side lengths. Prove that there is at least one among the small rectangles whose distances from the four sides of 
Prove that 
Y by PikaPika999
Let 
be a triangle and 
points in the same plane such that 
is the orthocenter of triangle 
, is the incenter of triangle 
, and is the centroid of triangle 
. If line 
is parallel to 
, prove that 
is the midpoint of segment .
be a triangle and 
points in the same plane such that 
is the orthocenter of triangle 
, is the incenter of triangle 
, and is the centroid of triangle 
. If line 
is parallel to 
, prove that 
is the midpoint of segment .
be inscribed in the unit circle so that
Since 
is the orthocenter of 
Since 
,
and so
And since 
bisects 
and 
bisects 
,
Since 
,
Then rearranging gives
as desired. 
