[/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.
.
to 
at points 
,
to 
is 
intersects line 
,
intersects line 
,
intersect at 
is tangent to 
.
,
,
is the point on 
.
be the circle through 
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 
and 
.
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 Adventure10 and 2 other users
In a trapezoid 
with a base 
, point 
is the orthogonal projection of 
on 
, and 
is the point on 
such that 
is perpendicular to . Let 
be the circumcenter of triangle 
. Suppose that the lines 
and 
have a common point. Prove that is a parallelogram.
with a base 
, point 
is the orthogonal projection of 
on 
, and 
is the point on 
such that 
is perpendicular to . Let 
be the circumcenter of triangle 
. Suppose that the lines 
and 
have a common point. Prove that is a parallelogram.
be the circumcircle of 
be the intersection of 
,
.
be the intersection of 
and 
is a diameter of 
but we knew that 
lie on a line,so 
.
so its sufficient to show that 
.
we get that 
so in triangle 
we get that 
.
s
we get the result...
,
,
so 
thus 
is cyclic.
