[/quote]
,
of positive integers such that
and 
,
be all positive integers smaller than 
such that 
for all 
is the largest positive integer that divides both 
and 
.
.
such that the sum of their cubes is equal to the square of their sum.
.
meets 
at points 
,
.
intersects line 
at 
,
intersects line 
,
intersect at 
is tangent to 
.
,
,
is the point on 
.
be the circle through 
tangent to 
at 
the circle through 
be the second intersection of 
lies on 
![$A_n=\left[n\right]^3{,}f\left(n\right)=\min \left|S\right|{,}S\ satisfy\ S\subseteq \ A_nand\ \forall \ x\in \ A_n{,}\exists \ y{,}z\in \ S\ {,}x{,}y{,}zcollinear$](http://latex.artofproblemsolving.com/a/a/6/aa671ae30c20a584500d6362790124ad6cbb67be.png)
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 
,
be the excircle opposite to 
and 
,
intersects line 
be the midpoint of 
is tangent to 
positive real numbers such that 
. Prove that :
such that for all 
,![\[
f(x + f(2y)) + f(x^2 - y) = f(f(x)) f(x + 1) + 2y - f(y).
\]](http://latex.artofproblemsolving.com/d/1/0/d10c01d7d416243614b24f04218d157e8b025ff2.png)
,
.
,
is acute, and let 
be its orthocentre. Points 
and 
,
and 
and 
and 
Y by
A quadrilateral 
is inscribed in a circle with a diameter of 
. Point 
is symmetrical to point 
wrt straight line , point 
is symmetric to point 
wrt straight line 
. Let 
be the point of intersection of lines 
and , 
be the point of intersection of lines 
and . Prove that 
.
is inscribed in a circle with a diameter of 
. Point 
is symmetrical to point 
wrt straight line , point 
is symmetric to point 
wrt straight line 
. Let 
be the point of intersection of lines 
and , 
be the point of intersection of lines 
and . Prove that 
.
be the symmetric of 
intersect at 
intersect at point of 
thus 
is on the polar of 
hence 
but since 
is the bisector of 
then 
