[/quote]
,
be an inscribed quadrilateral. Let the incenters of 
and 
be 
and 
respectively. Let the intersection point of the line that passes through 
and the line that passes through 
.
,
,
,![\[ \frac{a^3}{bc} + \frac{b^3}{ca} + \frac{c^3}{ab} \geq a+b+c \]](http://latex.artofproblemsolving.com/9/f/a/9fa6cca20b3620fbb32ee782196e96f6fffa0063.png)
satisfies the following condition.
,![$\sum_{k=1}^{n}\frac{1}{2}\left(1 - (-1)^{\left[\frac{n}{k}\right]}\right)a_k=1$](http://latex.artofproblemsolving.com/3/1/0/3109e3f024dc59c2cdad60efa270494d6095c330.png)
holds.
,
.
satisfy that 
.
![\[\left\lfloor\frac{2025}{1}\right\rfloor,\left\lfloor\frac{2025}{2}\right\rfloor,\left\lfloor\frac{2025}{3}\right\rfloor,\dots,\left\lfloor\frac{2025}{2024}\right\rfloor,\left\lfloor\frac{2025}{2025}\right\rfloor.\]](http://latex.artofproblemsolving.com/a/8/2/a8255ec218f896983d39044f8f777e2fd811479b.png)
,
,
.
and 
are located on 
and 
respectively, such that 
and 
.
to 
.
be the circumcircle of a triangle 
the midpoint of the arc 
.
and 
intersect the perpendicular bisectors of 
and 
,
intersect at 
.
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)
. The incircle 
is tangent to 
at 
respectively. A line through 
intersects 
be the second intersection of 
with 
intersects 
bisects 
![\[3\sqrt[3]{(2x^2-x-1)^2}-12\sqrt[3]{2x^2-x-1} +12-12x=x^3+2x^2-4x+1\]](http://latex.artofproblemsolving.com/c/9/c/c9cf0f481c4bc2ba25e47a1e8932d93a4edef325.png)
and 
be the circumcenter and orthocenter, respectively. Let 
and 
be the feet of the perpendiculars from 
and 
to the opposite sides, respectively. Show that if 
,
is perpendicular to 
.
Y by Adventure10, Mango247, lian_the_noob12, Rounak_iitr
Let 
be an isosceles triangle, in which 
, and let 
and 
be two points on the sides 
and 
, respectively such that 
. Suppose that the lines 
and 
intersects at 
. Prove that the bisectors of the angles 
and 
intersects at a point lying on the line
be an isosceles triangle, in which 
, and let 
and 
be two points on the sides 
and 
, respectively such that 
. Suppose that the lines 
and 
intersects at 
. Prove that the bisectors of the angles 
and 
intersects at a point lying on the line 
.
and 
so 
and 
are similar, we get 
and as a result, 
(1)
(Menelaus)
(because of (1))
Bisectors of 
intersect on 
,
,
of 
are concurrent at A-excenter of 
.
.
bisects 
e
,
lies on 
and 
(since 
is isosceles), 
,
,
.
bisects 
bisects if and only if 
.
, we get : 
and sine rule in 
gives us 
. Hence, the bisectors of the angles 
bisector of 
is the bisector of 
and 
is a bisector of 
is bisector of 
and 
and 
,
.
and 
.
, 
so 
we can see that 
.
is angle bisector of 
.
and 
we obtain 
and hence 
.
.
From this it follows that 
,
