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.
but I am looking for the exact value. Does anyone know how to solve this problem?
be a center of circle which passes through vertices of quadrilateral 
, which has perpendicular diagonals. Prove that sum of distances of point to sides of quadrilateral is equal to half of perimeter of .
of positive integers such that ![\[ k!=(2^n-1)(2^n-2)(2^n-4)\cdots(2^n-2^{n-1}). \]](http://latex.artofproblemsolving.com/3/5/3/353bbb15bac205053c26208af8cac7c32a296f55.png)
Proposed by Gabriel Chicas Reyes, El Salvador
cards on the table, and each card has a positive integer written on it. An odd number is written on exactly 
cards. Every minute, the following operation is performed: for all possible sets of 
cards on the table, the product of the numbers on these three cards is calculated, all the obtained results are summed, and this sum is written on a new card and placed on the table. A day later, it turns out that there is a card on the table, the number written on this card is divisible by 
Prove that one hour after the start of the process, there was a card on the table that the number written on that card is divisible by
on the board. Charlotte has two operations available: the GCD operation and the LCM operation.
and 
written on the board, erasing them, and writing the integer 
. and written on the board, erasing them, and writing the integer 
.
is called a winning number if there exists a sequence of operations such that, at the end, the only integer left on the board is . Find all winning integers among 
and, for each of them, determine the minimum number of GCD operations Charlotte must use. denotes the greatest common divisor of and , while the number denotes the least common multiple of and .
![\[ x^2 + 4 = y^5. \]](http://latex.artofproblemsolving.com/0/1/9/01984c4a7735ad128ca2c7c67f9c0fb312365291.png)
Bulgaria
and 
are in the coordinate plane such that 
. Let 
denote the locus of all points 
in the coordinate plane satisfying 
, and let 
be the midpoint of 
. Points 
and 
are on such that 
and 
. The value of 
can be expressed in the form 
, where 
and 
are relatively prime positive integers. Find 
.
knights and jokers are to be seated around a round table. They randomly pick the particular seats they want to sit at. What's the probability they'd be seated alternately?
, where 
is a prime number. Show that if is an even number, then the polynomial ![\[-1+\prod_{k=0}^{n-1} P\left(X^{p^k}\right)\]](http://latex.artofproblemsolving.com/5/d/f/5df80f885e63852e4f53a6145dafd18d63dc3526.png)
is divisible by 
.
are positive integers such that for every natural number the set of prime factors of 
are same. prove that 
.
and let 
be reals such that 
such that the number 
is a perfect square for all positive integers 
.
knight was already seated at the table so we don't have to take into account of the rotations if they were seated alternately. Though my solution does take into account of the different rotations of the knights and jokers.
