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 
divides 
.
If a problem has not been solved within 
days, then others are eligible to post a new probkem. Not only simple one-variable equations, but also systems of equations are allowed. The difficulty of these equations should be no less than that of typical quadratic one-variable equations. If the problem involves higher degrees or more variables, please ensure that the problem is solvable (i.e., has a definite solution, rather than an approximate one). Please indicate the domain of the solution to the equation (e.g., solve in 
, solve in 
).
be a scalene triangle, and let 
be a point on side 
satisfying 
. Suppose that 
and 
are points inside such that triangles 
and 
are similar and quadrilaterals 
and 
are cyclic. Let lines 
and 
meet at 
and lines 
and 
meet at 
. Prove that lines 
and 
are parallel.
which is a perfect square is of the form 
for 
be an integer. Prove that there exists a positive integer 
with the following property: For each positive integer , there is a prime number 
(possibly depending on 
) such that 
is divisible by , but not divisible by 
.
be a positive integer and let 
and 
be positive divisors of 
. Prove that 
is not divisible by 
.
and 
If each person remains there for 
minutes, what's the probability that all three people meet each other?
, denote the sum of digits of in its decimal representation by 
. Find all polynomials 
with integer coefficients such that for any positive integer 
, the integer 
is positive and 
is called 
if there exists a permutation 
of the numbers 
, such that:
and 
have different parities for every 
;
the sum 
is a quadratic residue modulo for every 
.
of positive integers with prime and ![\[ a^p=b!+p. \]](http://latex.artofproblemsolving.com/9/f/6/9f674dabcb364872cef4153198e2e362ae40a2fd.png)
the set of all positive integers. Find all functions 
such that for all positive integers 
and , the integer 
is nonzero and divides 
.
, there are exactly two or three positive real numbers satisfying the equation 
.
, 
. Points 
lie on segments 
respectively. Points 
are such that 
and 
are equlateral, 
and 
lie on opposite sides of 
, 
and 
lie on the opposite siedes of 
. Prove that
Find the maximum positive integer 
such that for every subset 
with n elements, there always exist two elements a, b in T such that:
and solve for the area of a hexagon in that 
square.
?