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.![\[\left\lfloor\frac{2025}{1}\right\rfloor,\left\lfloor\frac{2025}{2}\right\rfloor,\left\lfloor\frac{2025}{3}\right\rfloor,\dots,\left\lfloor\frac{2025}{2024}\right\rfloor,\left\lfloor\frac{2025}{2025}\right\rfloor.\]](http://latex.artofproblemsolving.com/a/8/2/a8255ec218f896983d39044f8f777e2fd811479b.png)
, 
, and 
. Points 
and 
are located on 
and 
respectively, such that 
and 
. Find the ratio of 
to 
.
be the circumcircle of a triangle 
. Denote by 
and 
the midpoints of the sides 
and 
, respectively, and denote by 
the midpoint of the arc 
of not containing 
. The circumcircles of the triangles 
and 
intersect the perpendicular bisectors of and at points 
and 
, respectively; assume that and lie inside the triangle . The lines 
and 
intersect at 
. Prove that 
.
which satisfy the following relationship for all real numbers 
and 
![\[f(xf(y)-f(x))=2f(x)+xy.\]](http://latex.artofproblemsolving.com/d/0/e/d0e9e56267aa6241a0323c7b2245e4752c338785.png)
be the set of nonzero real numbers. Find all functions 
such that, for all 
, ![\[(x-y)f(y^2)+f\left(xy\,f\left(\frac{x^2}{y}\right)\right)=f(y^2f(y)).\]](http://latex.artofproblemsolving.com/b/1/b/b1b1b156221baf8e9936027f352c50abcd2d05ff.png)
that is inscribed in 
. The incircle 
is tangent to 
at 
respectively. A line through parallel to intersects at , and 
intersects again at 
. Let is the midpoint of . 
be the second intersection of 
with . 
intersects again at . Prove that the line passing through perpendicular to 
bisects .
, let 
and 
be the circumcenter and orthocenter, respectively. Let and be the midpoints of and , respectively, and let 
and 
be the feet of the perpendiculars from 
and 
to the opposite sides, respectively. Show that if is the intersection of and 
, then 
is perpendicular to 
.
, 
, and 
be the midpoints of sides BC, CA, and AB, respectively. Point 
is located on the ray 
so that 
is similar to 
. Points 
and 
on rays 
and 
, respectively, are defined similarly. Prove that lines 
, 
, and 
are concurrent, i.e. these three lines intersect at a point.
be an acute triangle with circumcenter and orthocenter , satisfying 
. The tangent line at to the circumcicle of intersects in . Let be the midpoint of 
. Prove that 
.
less than three times of their sum. Find those integers.
be an acute-angled triangle 
with 
and incenter 
. Let 
be the mixtilinear circle at vertex 
, i.e. the circle internally tangent to the circumcircle of 
and also tangent to lines 
and 
. A circle 
passes through points 
and 
and is tangent to at point 
, with 
and being inside 
. Prove that:
![\[3\sqrt[3]{(2x^2-x-1)^2}-12\sqrt[3]{2x^2-x-1} +12-12x=x^3+2x^2-4x+1\]](http://latex.artofproblemsolving.com/c/9/c/c9cf0f481c4bc2ba25e47a1e8932d93a4edef325.png)
collinear points,consider the distances between the points.suppose each distance appears at most twice.prove that there are at least ![$[\frac{n}{2}]$](http://latex.artofproblemsolving.com/a/b/6/ab6ff27f204b4f6b5cb215e8324ea4d9da9b37ab.png)
distances that appear once each.
