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.
such that a positive integer power of 
equals to a positive integer power of 
.
be an acute-angled triangle with 
A circle passes through 
and 
and crosses the segments 
and 
again at 
and 
respectively. The circumcircles of 
and meet each other at points 
and 
The segments 
and 
intersect at 
Let 
and 
be the reflections of 
in the lines 
and 
respectively. Prove that the points 
and 
are concyclic.
and 
are primes and 
is a perfect square, prove that 
is a perfect square.
be positive real numbers such that 
. Prove that
When is the equality valid?
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
, 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 
.
, then you would submit 1734274).
, then you would submit 1734274).
.
from 
to 
,
from 
to 
.
is 30-60-90 and 
,
,
and the volume is 
,
.
.
,
.