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.
, and for all 
, define the sequence 
by the recurrence![\[
a_{n+1} = a_n^2 + 1
\]](http://latex.artofproblemsolving.com/3/1/1/3114595ded8c8961d611afaab9ad19e254973001.png)
Prove that there is no natural number 
such that![\[
\prod_{k=1}^{n} \left( a_k^2 + a_k + 1 \right)
\]](http://latex.artofproblemsolving.com/a/4/f/a4f88ee7e3d4e53e369319f74cd9b12be26c9681.png)
is a perfect square.
written on the board. We apply the following rule: At each step, we will add all the numbers that are the sum of the two numbers on the board so that the sum does not appear on the board. For example, if the two initial numbers are 
; then the numbers on the board after step 1 are 
; after step 2 are 
; 
, prove that the number 2024 cannot appear on the board.
; 
, prove that the number 2024 can appear on the board.
be a cyclic quadrilateral with circumcenter 
, such that 
is not a diameter of its circumcircle. The lines 
and 
intersect at point 
, so that 
lies between 
and , and 
lies between 
and . Suppose triangle 
is acute and let 
be its orthocenter. The points 
and 
on the lines and , respectively, are such that 
and 
. The line through , perpendicular to , intersects at 
, and the line through , perpendicular to , intersects at 
. Prove that the points , , are collinear.
be real numbers such that ![\[ab^2+bc^2+ca^2=6\sqrt{3}+ac^2+cb^2+ba^2.\]](http://latex.artofproblemsolving.com/3/d/0/3d015292c249a69a44d5f923092201d7a271b896.png)
Find the smallest possible value of 
.
distinct positive integers and then writes in blue the 
sums of each two red numbers. At most how many of the blue numbers can be prime?
, let a line 
pass through the orthocenter and not through point . The line intersects line at 
. A line passing through and perpendicular to meets the circumcircle of triangle at 
. Let the feet of the perpendiculars from 
to be 
, respectively. Define line 
as the line passing through 
and perpendicular to , and line 
as the line passing through 
and perpendicular to 
. Prove that if 
is the reflection of the intersection of and across , then 
.
, let 
be the number of positive integer divisors of .
we have that:![\[ d(a)+d(b) \le d(\gcd(a,b))+d(\text{lcm}(a,b)) \]](http://latex.artofproblemsolving.com/a/3/b/a3bfe462d3888517939eec8a2827ee6bd8d627a1.png)
and determine all pairs of positive integers where we have equality case.
denote the set of positive integers. Find all functions 
such that for positive integers 
and 
![\[f(a^2 + b^2) = f(a)f(b) \text{ and } f(a^2) = f(a)^2.\]](http://latex.artofproblemsolving.com/c/1/f/c1f8ffe04cfcc45b498f3931a4796d5c56dc04d0.png)
. A convex polygon is said to be -good if it contains lattice points where any three of them are not collinear.-good convex polygon with area at most 
.
so that any -good convex polygon has area at least 
.
and 
be concentric circles, with in the interior of . From a point on one draws the tangent 
to (
). Let be the second point of intersection of and , and let be the midpoint of . A line passing through intersects at and in such a way that the perpendicular bisectors of 
and 
intersect at a point 
on . Find, with proof, the ratio 
.
be the orthocenter of acute triangle , let be the foot of the altitude from to , and let be the reflection of across . Suppose that the circumcircle of triangle 
intersects line at two distinct points 
and 
. Prove that is the midpoint of 
.
be a quadrilateral inscribed in circle 
with 
. Let and be the reflections of over lines 
and , respectively, and let be the intersection of lines 
and 
. Suppose that the circumcircle of 
meets at and , and the circumcircle of 
meets at and 
. Show that 
.
.