[/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 
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 
and 
.
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
Given is a triangle 
and a point 
within the triangle. The point is mirrored in the sides of the triangle: 
and 
are the mirrorings of in 
and 
, respectively . 
is the center of the circle passing through the vertices of triangle 
. is mirrored again in the sides of triangle : 
and 
are the mirror of in 
respectively, 
and .
a. Prove that is the center of the circle passing through the vertices of triangle 
.
b. Where should you choose within triangle so that and coincide? Prove your answer.
and a point 
within the triangle. The point is mirrored in the sides of the triangle: 
and 
are the mirrorings of in 
and 
, respectively . 
is the center of the circle passing through the vertices of triangle 
. is mirrored again in the sides of triangle : 
and 
are the mirror of in 
respectively, 
and .a. Prove that
is the center of the circle passing through the vertices of triangle 
.b. Where should you choose
within triangle so that and coincide? Prove your answer.This post has been edited 1 time. Last edited by parmenides51, Jan 29, 2020, 11:23 PM
are isogonal conjugates in triangle 
are respectively symmetric with respect to 
.
.
we get 
.
