[/quote]
,
,
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 
,
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 
.
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 
pairwise distinct complex numbers are written on a board. It turns out that there are exactly 760 ways to choose two numbers 
and 
from the board such that:![\[
a^2 + b^2 + 1 = 2ab
\]](http://latex.artofproblemsolving.com/8/f/c/8fc5df8dbc8e1647edfaa96dcfa8a0dfdac0c961.png)
Ways that differ by the order of selection are considered the same. Prove that there exist two numbers 
and 
from the board such that:![\[
c^2 + d^2 + 2025 = 2cd
\]](http://latex.artofproblemsolving.com/2/7/0/2706d72797c5e392b3f068a8174bd257e4d93fa1.png)
,
are even?
Y by centslordm, HWenslawski
Let 
be a circle, 
be a point outside it, and 
and 
the intersection points between and the tangents from to . Let 
be a point on the line 
, distinct from and and let 
be the second intersection point of and the circumcircle of the triangle 
.Also, let 
be the reflection of in point .
Show that
be a circle, 
be a point outside it, and 
and 
the intersection points between and the tangents from to . Let 
be a point on the line 
, distinct from and and let 
be the second intersection point of and the circumcircle of the triangle 
.Also, let 
be the reflection of in point .Show that
![[asy]
size(300);
pair P,A,B,K,O,O1,T,P1;
P=(0,0);
A=dir(-35);
B=dir(35);
K=arcpoint(A--B,5/7*abs(A-B));
O=2*circumcenter(P,A,B);
O1=circumcenter(P,K,B);
T=reflect(O,O1)*B;
P1=2*A;
draw(circle(O,abs(A-O)));
draw(circumcircle(P,B,T));
draw(P--B);
draw(T--K);
draw(B--A--T--cycle);
draw(K--P1--A);
draw(P--T--A--cycle);
label("$P$",P,dir(200));
label("$A$",A,SW);
label("$B$",B,dir(70));
label("$K$",K,E);
label("$T$",T,dir(-120));
label("$P'$",P1,SE);
[/asy]](http://latex.artofproblemsolving.com/d/4/7/d476a3a46794f956bd18efd23a6ffee9e3da9b91.png)
is tangent to 
.
,
tangent 
,
.
,
so 
.
Thus by SAS we have 
,
.
,
,
.
,
.
and 
are similar.
and 
.
and 
are similar.
and 
.
