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.
is chosen. Prove that there exists a prime number that divides infinitely many terms of the sequence 
, where![\[
b_k = a^{k^k} \cdot 2^{2^k - k} + 1.
\]](http://latex.artofproblemsolving.com/1/7/5/1751a60482d729a36c71b77ac9c978e724f40da0.png)
is polynomial with minimum degree such that 
for 
, find 
such that 
and 
.
be a triangle and let its incircle, centred at 
, touches the side 
at 
. A line through 
intersects the lines , 
and 
at 
, 
and 
, respectively. The circle 
intersects the circles 
and 
again at 
and 
, respectively. Prove that the points , , and are concyclic.
of natural numbers such that
ones are written in a circle. Petya and Vasya take turns making 
moves each. In each move, Petya chooses 9 consecutive numbers and decreases each by 
. Vasya chooses 
consecutive numbers and increases each by 
. They alternate turns, starting with Petya. Prove that Vasya can act in such a way that after each of his moves, there are always at least five positive numbers, regardless of how Petya plays. be an acute-angled triangle with circumcircle 
and circumcentre 
. Points 
and 
lie on such that 
and 
. Let 
meet 
at , and 
meet 
at .
and 
have an intersection lie on line 
.
with three fixed points 
and 
on it. Also, let us fix a real number 
. For a variable point 
on , let 
be the point on the segment 
such that 
. Let 
be the second point of intersection of the circumcircles of the triangles 
and 
. Prove that as 
varies, the point lies on a fixed circle.
be a positive integer and let 
be an irrational number in 
. Show that for every integer 
, there exists a positive integer 
such that 
.
is a sequence of real numbers with 
such that for all 
Prove that
, let be a point in its interior, and let 
be a line through . Construct with proof using ruler and compass, all circles which pass through , are tangent to and whose center lies on line .
be the line through 
to .
.
and 
be the 2 midpoints of arc of .
and let 
and 
.
and 
.
and then construct the tangent to which is perpendicular to . Let this tangent meet at . Then the required circle is one with diameter as and the inverse of .
denote the map inversion of the circle center at a point. ,
is drawable with rular and compass.
. be our needed circle.
is a line perpendicular to the line 
.which touch the circle..
such that 
Prove that 
Where 
Prove that :
Where 
then find minimal value of 
Find the value of ![$[100A] .$](http://latex.artofproblemsolving.com/f/5/c/f5cb78c5a80d7eb64d2887c195a92ba19375626f.png)
