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 an acute triangle and let 
be an interior point for it such that 
. Prove that
When does equality hold?
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 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 
.
such that for all 
,![\[f(x+yf(x))+y = xy + f(x+y).\]](http://latex.artofproblemsolving.com/c/f/3/cf3d20a041c27244e90876119b4568b7a3e13c03.png)
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 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.
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.
be a positive integer. Prove that there exists a positive integer and distinct primes 
such that if 
denotes the number of positive integers less than or equal to and not divisible by any of , then
be an integer and denote 
. For a function 
we denote Im 
, Fix 
and 
. Find all possible values of 
Im 
+ Fix + 
.
with 
.![\[
(x^{2}-yz)(y^{2}-zx)(z^{2}-xy).
\]](http://latex.artofproblemsolving.com/a/8/7/a87fae27584012a867bc0a90190c0317f9f7ab07.png)
so that:
with all real numbers 
satisfies 
satisfies the equality 
. Let be the center of the circumscribed circle of triangle , be the midpoint of the edge . Prove that lines and 
are perpendicular.
satisfies the equality . Let be the center of the circumscribed circle of triangle , be the midpoint of the edge . Prove that lines and are perpendicular.
then 
varies along a circle 
around the axis 
. But then 
varies along the circle 
obtained from by homothety with center 
and coefficient 
. The plane of is obviously perpendicular to . Thus it suffices to prove that 
where 
is the midpoint of 
and 
is the intersection of and and 
lie in the same side of .. Draw the lines 
and 
through 
and 
, respectively, perpendicular to . Then and are isogonal conjugates in the degenerated triangle formed by 
. Thus is the center of the circumcircle of the rectangle formed by the projections of on 
. This is possible only if lies on the midline of , i.e. the perpendicular bisector of . Thus 
.
