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.
be a quadrilateral inscribed in a circle of center 
. Let M and N be the midpoints of diagonals 
and 
, respectively and let 
be the intersection point of the diagonals and of the given quadrilateral .It is known that the points 
are distinct. Prove that the points 
are concyclic if and only if the points 
are concyclic.
such that there exists a pair 
of positive integers, such that 
is not divisible by the cube of any prime, and 
such that has exactly 2000 prime divisors and divides 
?
be triangle with 
and 
on the segment 
such that 
. Consider point 
on 
such that 
. Point 
is the intersection of tangent line through point to circle 
respectively . If the tangent through point to 
intersect 
at 
. Line 
intersect 
at 
. Prove that 
bisect 
.
of positive integers satisfying the equation![\[ x!+y!=x^{y}.\]](http://latex.artofproblemsolving.com/e/6/c/e6c7458015346f21922d375340c4ba142afbdce5.png)
be a positive integer and consider the set 
.
of numbers not previously chosen such that for every distinct 
, neither 
divides 
nor divides .
, all multiples of every element in , including those in itself, are removed from 
. until no moves are possible., which player has a winning strategy if any
be a triangle and 
an interior point of . Show that at least one of the angles 
is less than or equal to 
.
. Let 
be the midpoint of the arc 
not containing 
and define 
similarly. Show that the orthocenter of 
is the incenter 
of .
table filled with natural numbers, we say it is a divisor table if:
-th row are exactly all the divisors of some natural number 
,
-th column are exactly all the divisors of some natural number 
,
for every 
.
is given. Determine the smallest natural number 
, divisible by , such that there exists an divisor table, or prove that such does not exist.
such that 
,![\[
f(x^2+f(y))=f(xf(x))+y
\]](http://latex.artofproblemsolving.com/d/e/d/dedd2338779fcc6723077ce586713fb4a68b9931.png)
be a triangle with incenter and let be an arbitrary point on the side . Let the line through perpendicular to 
intersect 
at 
. Let the line through perpendicular to intersect at 
. Prove that the reflection of across the line 
lies on the line .
be an acute triangle. Let 
be the reflection of point over the line . Let 
and 
be the circumcenter and the orthocenter of triangle , respectively, and let be the midpoint of segment 
. Let and 
be the points where the reflection of line 
with respect to line 
intersects the circumcircle of triangle , where point lies on the arc not containing . If 
is a point on the line 
such that 
, prove that 
.
Let 
and 
is the sum of 
square numbers. Prove that: 
is the sum of square numbers
Let and is the sum of square numbers. Prove that: is the sum of square numbers
Let and is the sum of square numbers. Prove that: is the sum of square numbers
be equal to 
divided by the largest power of 
that divides . Legendre's three-square theorem says that is expressible as the sum of three squares iff is not 
mod 
.
is never mod , so even always works. If is odd, if 
is mod , then 
is even and thus 
is even. Thus 
is odd. Observe that if is odd then 
. Thus if is mod then 
is mod . However, any odd number to an odd power is congruent to itself mod , so this means 
, contradiction.
, which would make it trivial.
such that 
.
