be a positive integer. Prove that for every odd prime 
dividing 
,
such that 
.
be an acute triangle with incentre 
and 
.
and 
intersect the circumcircle of 
and 
,
and 
such that 
and 
are parallelograms (with 
,
)
be the point of intersection of lines 
and 
.
,
that satisfy![\[f(xf(y) + f(x)) = f(x)f(y) + 2f(x) + f(y) - 1,\]](http://latex.artofproblemsolving.com/4/7/f/47f094ce57fe35d339ec3b93152ec3459a22ec48.png)
for every 
,
for every 
and 
,
.
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 
Y by
Let a quadrilateral 
have an inscribed circle and let 
be the tangency points of the sides 
and 
, respectively. Consider the orthocenters of each of the triangles 
and 
. Prove that these four points are the vertices of a parallelogram.
have an inscribed circle and let 
be the tangency points of the sides 
and 
, respectively. Consider the orthocenters of each of the triangles 
and 
. Prove that these four points are the vertices of a parallelogram.This post has been edited 1 time. Last edited by parmenides51, Jun 11, 2020, 4:52 AM
Reason: name typo
Reason: name typo
