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.
vertices is given. To each vertex, a natural number from the set 
is assigned. Prove that there are 
vertices 
such that if the numbers 
are assigned to them respectively, then 
and 
is a parallelogram.
be a triangle inscribed in a circle 
with orthocenter 
and altitude 
. Let 
be the midpoint of 
. Line 
meets again at 
. Line 
meets again at 
. Let 
be the orthogonal projection of 
on the line 
. Line 
meets again at 
. Prove that 
.
consecutive even numbers, such that the sum of their squares divides the square of their product.
satisfying 
, find the biggest value the following equation could acquire:
and 
on 
and 
respectively in 
such that 
then draw line 
through 
such that 
let 
and 
reflection of 
and to respectively prove that 
and are congruence.
be a set of 
distinct integers all of whose prime factors are at most 
Prove that contains 
distinct integers such that their product is a perfect cube.
such that
for all 
.
and 
are natural numbers such that
is a prime number. Find all possible values of ,,
.
is given. Let 
and 
be the midpoints of and , respectively. on segment 
is given. Draw equilateral 
such that is on and is on 
.
and all real numbers 
:
, 
be the orthocenter of it and 
be any point on the side 
. The points 
are on the segments 
, respectively, such that the points 
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 
. Prove that the points 
and lie on the same line.
be a triangle with 
and 
. and are marked on and respectively, such that 
. Segments 
and 
intersect at . Find 
and 
.
