AoPS Academy
be a scalene triangle with point 
and 
on the plane such that 
is an equilateral . Let 
intersect 
and 
at 
and 
respectively and 
intersect 
and 
at 
and 
respectively .
be triangle with point 
and 
on 
and 
are on the same side of 
be midpoint of segment and 
be midpoint of segment 
be intersection of 
with 
and 
with incenter 
,
is a point on ,the bisector of 
meet 
at .Prove that 
such that
for all positive real numbers 
and 
.
Prove that: 
is the sum of 
square numbers
Let 
and 
is the sum of square numbers. Prove that: 
is the sum of square numbers
real number on the board such that each number is more than 
.
on the board and write 
.
and 
.
?
be triangle and point be 
altitude of . be a point on and such that 
and 
. Point is the intersection of 
and 
. Point be intersection of 
and . Line 
intersect circumcircle of again at .
.
bisect .
, define a sequence of integers 
by letting 
and 
for 
. Let 
. Determine the largest possible 
such that, for some positive integer , the numbers 
are all prime.
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 as a sequence of distinct cities 
, 
, such that there are direct flights between 
and 
for every 
;
A 
between and as a path between and such that no other path between and has more cities;
A 
between and as a path between and such that no other path between and has fewer cities. and in the country, there exist a long path and a short path between them that have no cities in common (except and ). Let be the total number of pairs of cities in the country that are connected by direct flights. In terms of , find all possible values 
) and you also need to be quite good at algebra. You should start with basic euclidean though and perhaps learn inversion first before trying bashing, you will build the fundamentals you need that way.
a
.
