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.
are written on a board. 
and 
plays the following game: They take turns choosing a number from the board and deleting them. starts first. They sum all the deleted numbers. If after a player's turn (after he deletes a number on the board) the sum of the deleted numbers can't be expressed as difference of two perfect squares,then he loses, if not, then the game continues as usual. Which player got a winning strategy?
be an acute triangle with 
. be 
the circumcircle circumscribed to the triangle and 
the midpoint of the smallest arc 
of this circle. Let 
and 
points of the segments 
and 
respectively such that 
. Let 
be the second intersection point of the circumcircle circumscribed to 
with . Let 
and 
be the intersections of lines 
and 
with other than 
, respectively. Let 
and 
be the intersection points of lines 
and 
with lines and respectively. Show that the 
line passes through the midpoint of
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?
be nonzero real numbers such that 
. Assume that 
Show that for any odd integer 
, 
satisfy the equality, 
prove that the triangle must have a right angle.
denote the set of natural numbers, and let 
, 
, be nine distinct tuples in 
. Show that there are three distinct elements in the set 
whose product is a perfect cube.
and 
be prime numbers. Prove that if 
is a perfect square, then 
is also a perfect square.
points in 
can be partitioned into two sets whose convex hulls intersect.
and 
and 
be six positive real numbers. Prove that
