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.
is not divisible by 
for any positive integer 
.
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 
.
be an acute-angled triangle with 
, orthocenter 
, circumcircle 
and circumcentre 
. Let 
be the midpoint of 
and let 
be a point such that 
is a parallellogram. Suppose that there exists a point 
on and on the opposite side of 
to 
such that 
. Let 
be the midpoint of 
. Prove that if 
, then 
.
, 
be the orthocenter of it and 
be any point on the side 
. The points 
are on the segments 
, respectively, such that the points 
and 
are cyclic. The segments 
and 
intersect at 
is a point on 
such that 
is tangent to the circumcircle of triangle 
at 
and 
intersect at 
. Prove that the points 
and lie on the same line.
such that for all 
,![\[f(x+yf(x))+y = xy + f(x+y).\]](http://latex.artofproblemsolving.com/c/f/3/cf3d20a041c27244e90876119b4568b7a3e13c03.png)
be the set of all lines in the plane and be a set of 
points in the plane. For a line 
let 
be the number of distinct points among the orthogonal projections of the points from on 
. Find the maximum possible number of distinct values of (this quantity is computed for any line ) as varies.
, where 
, the internal angle bisectors of angles 
, 
, and 
meet the sides , 
, and 
at points , 
, and 
, respectively. Let 
be the incenter of triangle 
, and let 
be the foot of the perpendicular from to line 
. Prove that if the quadrilateral 
is cyclic, then the center of its circumcircle lies on segment 
.
be an acute triangle and let 
be an interior point for it such that 
. Prove that
When does equality hold?
be an acute scalene triangle , be the orthogonal projection of on , and 
are the midpoints of 
and 
respectively. Let and 
are points on the minor arcs 
and 
of the circumcircle of triangle respectively such that 
. Show that the circumcircles of triangles 
and 
are tangent if and only if lies on 
.
be a triangle and the points 
and 
on , and on , and and on be such that 
and 
. The intersections of 
with 
and 
are 
and 
respectively, and the intersections of 
with 
and 
are and 
, respectively. Prove that the lines 
and 
are concurrent.
be fixed points on this order on a line. Let 
be a variable circle through 
and and suppose it meets the perpendicular bisector of 
at the points and . Let 
and be the other points of intersection of 
and 
with . Prove that 
passes through a fixed point independent of .
, such that 
, there exists an integer satisfying 
?
the sum of all positive divisors of .
be an integer. Alice and Bob play the following game on the vertices of a regular -gon. Alice places her token on a vertex of the n-gon. Afterwards Bob places his token on another vertex of the n-gon. Then, with Alice playing first, they move their tokens alternately as follows for 
rounds: In Alice’s turn on the 
-th round, she moves her token positions clockwise or anticlockwise. In Bob’s turn on the -th round, he moves his token 
position clockwise or anticlockwise. If at the end of any person’s turn the two tokens are on the same vertex, then Alice wins the game, otherwise Bob wins. Decide for each value of which player has a winning strategy.
be an integer. Prove that 
is not a perfect cube of an integer.
are two circle which circumscribed and tangjent to each other . If we write with 
and the radius of smaller and bigger circle respectively and if 
find .
be the centers of the circles (small and big, respectively). Let 
be the perpendiculars to the side 
, and 
the perpendicular to 
. Then 
, 
, and 
since 
is the bisector of 
. Apply the Pythagorean theorem to the right triangle 
to get 
