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.
satisfies 
. Prove that 
.
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 
.
be real numbers such that 
. Find the smallest possible value of 
.
into 
or 
can sometimes be quite complex. That's why I’ve written a program to assist with this process.
such that for all positive real numbers 
and 
, the following equation holds:![\[
\frac{x + f(y)}{x f(y)} = f\left( \frac{1}{y} + f\left( \frac{1}{x} \right) \right)
\]](http://latex.artofproblemsolving.com/7/d/f/7df25755bc33bef970fffb392fb5b6b789390881.png)
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.
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?
be a triangle. Points and lie on sides and 
, respectively, such that 
. Segments and 
meet at . The bisector of angle 
meet segments and at 
and 
, respectively. Prove that 
.
and 
are the -angle bisector of the 
and 
.So applying angle bisector theorem we get,![\[\frac{QD}{QA}=\frac{CD}{AC} \implies \frac{AD}{QA}=\frac{CD+AC}{AC}=\frac{CD+CE+AE}{AC}.\]](http://latex.artofproblemsolving.com/f/5/c/f5c19abc34178a2dcace41939ab72a769a95aa7d.png)
![\[\frac{EC}{ER}=\frac{BD+EC+CD}{BE}\]](http://latex.artofproblemsolving.com/9/4/f/94ffeba8d2a23e2f2f0915f061cefa410083844e.png)
.Cancelling two common terms from these two equations we get, ![\[\frac{QA}{AC}\cdot \frac{EC}{ER}=\frac{AD}{BE}\]](http://latex.artofproblemsolving.com/0/8/c/08cdc68f9007dffc0c29d7c7737644f857470657.png)
with the transversal 
we get,![\[ \frac{ER}{PR}\cdot \frac{AC}{EC}\cdot \frac{PQ}{QA}= 1 \implies \frac{PQ}{AD}=\frac{PR}{BE}\]](http://latex.artofproblemsolving.com/1/7/6/1766bc54c3dec573f12fd027e9bdafaa97efcd81.png)
And since 
we have:
Now using Menelaus Theorem in 
wrt 
