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
![\[x^5 + ax^4 + bx^3 + cx^2 + dx + e = 0\]](http://latex.artofproblemsolving.com/f/d/0/fd0ab11635ee44ad272a54c0f20a2bc47a3596de.png)
cannot all be real if 
.
not necessarily distinct factorials of positive integer numbers. Prove that the set of compact powers of two is finite.
is called lucky if it has at least two distinct prime divisors and can be written in the form:![\[
n = p_1^{\alpha_1} + \cdots + p_k^{\alpha_k}
\]](http://latex.artofproblemsolving.com/7/4/4/744a5ccaeb9476ebd7d999c395762cb6e99a7a71.png)
where 
are distinct prime numbers that divide . (Note: it is possible that has other prime divisors not among .) Prove that for every prime number 
, there exists a lucky number such that 
.
be a triangle. A circle passing through 
and 
intersects segments 
and 
at 
and 
, respectively. Lines 
and 
intersect at 
, while lines 
and 
intersect at 
. Prove that 
if and only if 
.
satisfy 
. Determine the minimal and the maximal value of the expression 
are positive integers, and smallest side is equal to 
. Determine the area of triangle 
if 
, where 
, 
and 
are lengths of altitudes in triangle from vertices , and 
, respectively.
be a triangle and let 
be its incircle. Denote by 
and 
the points where is tangent to sides and , respectively. Denote by 
and 
the points on sides and , respectively, such that 
and 
, and denote by the point of intersection of segments 
and 
. Circle intersects segment at two points, the closer of which to the vertex is denoted by 
. Prove that 
.
, 
be positive integers such that 
is a multiple of 
for all positive integers 
. Prove that 
.
be an isosceles triangle with vertex in and 
. Let be the foot of the bisector from . Show that 
.
the following equation :
