AoPS Academy
[/quote]
, do not answer with "
is a solution" only. Either you post any kind of proof or at least something unexpected (like "
is the smallest solution). Someone that does not see that is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
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 .
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 
such that 
for all real number 
and 
be a sequence of perfect squares such that 
can be obtained by concatenating a digit to the right of 
. Determine all such sequences that are of maximum length.
, 
is cyclic, and it's circumcenter is the midpoint of 
, now 
, then 
and 
is cyclic so 
a
.
. Prove that
be a quadrilateral inscribed in a circle. Point 
satisfies 
.
passes through the circumcenter of triangle 
.
