[/quote]
,
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 a finite set, and let 
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
be the set of integers. Determine all functions 
such that, for all integers 
and 
,
Proposed by Liam Baker, South Africa
that makes the following equation have no integer root 
.
Y by Adventure10
The point 
is marked on the segment 
of the triangle 
. Let 
, 
, 
be the centers of the inscribed circles 
, 
, 
of the triangles , 
, 
, respectively. A common external tangent to the circles and , different from the line , intersects the segment 
at the point 
. It is known that the points and do not coincide and lie inside 
. Prove that in the triangle 
the points and are isogonally conjugate. (The interior points 
and 
of the triangle are called isogonally conjugate if 
, 
, 
.)
is marked on the segment 
of the triangle 
. Let 
, 
, 
be the centers of the inscribed circles 
, 
, 
of the triangles , 
, 
, respectively. A common external tangent to the circles and , different from the line , intersects the segment 
at the point 
. It is known that the points and do not coincide and lie inside 
. Prove that in the triangle 
the points and are isogonally conjugate. (The interior points 
and 
of the triangle are called isogonally conjugate if 
, 
, 
.)
if 
are pairs of isogonal conjugates then 
.
and 
are isogonal pairs WRT 
.
be the pairs of Isogonal Conjugate WRT 
.
.
.
also 
.
