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.
cards on a table, flipped face down. Madina knows that on each card a single number is written and that the numbers are different integers from 
to . In a move, Madina is allowed to choose any 
cards, and she is told a number that is written on one of the chosen cards, but not which specific card it is on. After several moves, Madina must determine the written numbers on as many cards as possible. What is the maximum number of cards Madina can ensure to determine?
, 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.
+1 w
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.
. 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 
.
with 
and let be 
it’s circumcircle. Diameters 
and 
are drawn. Circle 
interescts 
at 
. Circle 
intersects 
at 
.(
lies between 
and ). Prove that lines 
and 
intersect at circle 
.
has been partitioned into disjoint pairs 
(
) so that for all 
, 
equals or 
. Prove that the sum ![\[ |a_1-b_1|+|a_2-b_2|+\cdots +|a_{999}-b_{999}| \]](http://latex.artofproblemsolving.com/0/4/4/04422a7d2266b92a6b3f215b698f078cda85091f.png)
ends in the digit 
.
is inscribed in the circle. Construct a point such that the points of intersection of lines 
and 
with this circle are the vertices of an equilateral triangle.
is inscribed in the circle. Construct a point such that the points of intersection of lines and with this circle are the vertices of an equilateral triangle.
and such that for any point of the circle which lies on the same side of the plane as with respect to 
. Consider an analogous circle passing through and . Let be the point of their intersection. Then ; 
; 
. It's not hard to see that this point suffices the conditions
