[/quote]
,
such that for all 
,![\[f(x+yf(x))+y = xy + f(x+y).\]](http://latex.artofproblemsolving.com/c/f/3/cf3d20a041c27244e90876119b4568b7a3e13c03.png)
is called fresh if there exists no positive integer 
such that the first 
numbers in the permutation are 
in some order. Let 
be the number of fresh permutations of the integers 
for all 
.
,
is fresh, whereas the permutation 
is not.
cities in a country, where 
is an integer. Some pairs of cities are connected by direct (two-way) flights. For two cities 
and 
we define:
A 
between 
,
,
and 
for every 
;
A 
between 
A 
between 
be the total number of pairs of cities in the country that are connected by direct flights. In terms of 
,
be the orthocenter of it and 
be any point on the side 
.
are on the segments 
,
and 
are cyclic. The segments 
and 
intersect at 
is a point on 
such that 
is tangent to the circumcircle of triangle 
at 
and 
intersect at 
.
and 
be the set of all functions from 
be an element of 
be the image of 
for every 
in 
.
.
and 
intersect each other at points 
and 
.
be the reflection of 
.
.
satisfying :![\[n^{2p}=m^2+n^2+p+1\]](http://latex.artofproblemsolving.com/7/4/e/74ee592cb8b3f3a3996ea34f40b78fde9d3672fc.png)
and 
is a prime number.
Find all the integers 
be a triangle with incentre 
and circumcircle 
.
and 
be the second intersection points of 
and 
,
meets 
at a point 
at a point 
.
and the line through 
.
.
and 
are either parallel or concurrent.
Prove that
is an odd number, let 
be a positive integer. If 
then 
satisfying
and 
Prove that
Y by Adventure10
Let 
be points on the circumference of a given circle 
. From the triangle 
, called 
, the triangle 
is obtained by constructing the points 
on such that 
is parallel to 
, 
is parallel to 
, and 
is parallel to 
. Each angle of 
is an integer number of degrees and those integers are not multiples of 
. Prove that at least two of the triangles 
are congruent.
be points on the circumference of a given circle 
. From the triangle 
, called 
, the triangle 
is obtained by constructing the points 
on such that 
is parallel to 
, 
is parallel to 
, and 
is parallel to 
. Each angle of 
is an integer number of degrees and those integers are not multiples of 
. Prove that at least two of the triangles 
are congruent.
