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.
such that
for all positive real numbers 
and 
.
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.
must be in the interior of . In other case as is convex, we can take a line passing through and not intersecting . So let be our point in the interior of quadrilateral , which we will denote with 
. Let 
be a line passing through and intersecting and 
in points and respectively. Let 
and 
be such points on that 
and 
. Let lines 
and 
intersect side in 
,
respectively. By 
we will denote area of triangle 
and the same thing for quadrilaterals... As lines 
,
and 
are lines dividing into two parts of equal area, we obtain equations ![\[ S(KQX)+S(XPL)=S(QMX)+S(NPX) \]](http://latex.artofproblemsolving.com/1/4/e/14e11850bd3579356df0fff5d47457b2060135b8.png)
but as 
we have 
which means that 
.Now let be a line passing through and parallel to this line intersects with lines 
and in points 
and 
respectively. As we have we have also 
,but there is also equality , and so on, if 
it would mean that 
which is impossible. We obtained that 
, so 
.
which means that is a parallelogram! 
Now , if is a parallelogram for point we can take intersection point of it's diagonals 
a
.
. Prove that
