[/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 
.
intersects line 
,
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 
![$A_n=\left[n\right]^3{,}f\left(n\right)=\min \left|S\right|{,}S\ satisfy\ S\subseteq \ A_nand\ \forall \ x\in \ A_n{,}\exists \ y{,}z\in \ S\ {,}x{,}y{,}zcollinear$](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 
and 
,
and 
and 
and 
Y by
Let 
be a convex cyclic quadrilateral with non-parallel opposite sides such that 
is a diameter of the circumcircle of . Lines 
and 
meet at 
. Lines 
and 
meet at 
. The internal bisector of 
meets and at 
and 
, respectively. The internal bisector of 
meets and at 
and 
, respectively.
Prove that the circumcircles of the triangles
and 
are tangent.
be a convex cyclic quadrilateral with non-parallel opposite sides such that 
is a diameter of the circumcircle of . Lines 
and 
meet at 
. Lines 
and 
meet at 
. The internal bisector of 
meets and at 
and 
, respectively. The internal bisector of 
meets and at 
and 
, respectively.Prove that the circumcircles of the triangles
and 
are tangent.This post has been edited 1 time. Last edited by parmenides51, Sep 6, 2021, 8:39 PM
