be the side lengths of a triangle. Prove that![\[\frac{ab+bc+ca}{(a+b)^2} + \frac{ab+bc+ca}{(b+c)^2} + \frac{ab+bc+ca}{(c+a)^2} \leqslant \frac{85}{36}.\]](http://latex.artofproblemsolving.com/f/8/0/f80aed170f4f4bd78a2e881ce93b6b7ddfa03c22.png)
such that the reflection of each of these four points about 
,![\[
\frac{ab}{b+1} + \frac{bc}{c+1} + \frac{ca}{a+1} \geq \frac{3abc}{1 + abc}
\]](http://latex.artofproblemsolving.com/d/5/8/d582a3269c92041834c02153ee8e946a274e52bd.png)
be a triangle with 
and 
.
inside triangle ![\[
XA^2 + BC^2 \leq XC^2 + AB^2 \leq XB^2 + AC^2.
\]](http://latex.artofproblemsolving.com/0/3/1/031aa58d7d3dabe636dfb4245ef2456a2a3d5df6.png)
Find the maximum value of 
such that![\[
\frac{XA}{AC} \geq m.
\]](http://latex.artofproblemsolving.com/c/5/e/c5ed2892b224aff3b2f3e20b4c9424707ff88690.png)
The diagonals 
at 
The line through 
mmets the extension of the side 
beyond 
at 
The line through 
meets the extension of the side 
Prove that the circumcircles of the triangles 
and 
are tangent .
be non-negative real numbers such that 
Prove that
board. On each cell of the main diagonal there is a file. At each step, we may select some files and move them to the left, on their respective rows, by the same distance. What is the minimum number of necessary moves in order to put all files on the first column?
and 
. Prove that
Let 
be reals such that 
. Prove that
![\[\frac{a^3}{(a+b)^3}+\frac{b^3}{(b+c)^3}+\frac{c^3}{(c+a)^3} \leqslant \frac{9}{8}.\]](http://latex.artofproblemsolving.com/5/1/5/515cbd3cd69fee483312f90d4cb29d98ca761f41.png)
Y by sixoneeight, OronSH, goaoat
Quadrilateral 
has an incenter 
Suppose 
. Let 
be the midpoint of 
. Suppose that 
. 
meets 
again at point 
. Let points 
and 
be such that is the midpoint of 
and is the midpoint of 
. Point 
lies on the plane such that 
is a parallelogram, and suppose the angle bisectors of 
and 
concur on 
.
The angle bisectors of
and 
meet 
at 
and 
. Prove that 
.
has an incenter 
Suppose 
. Let 
be the midpoint of 
. Suppose that 
. 
meets 
again at point 
. Let points 
and 
be such that is the midpoint of 
and is the midpoint of 
. Point 
lies on the plane such that 
is a parallelogram, and suppose the angle bisectors of 
and 
concur on 
.The angle bisectors of
and 
meet 
at 
and 
. Prove that 
.This post has been edited 1 time. Last edited by popop614, Apr 3, 2025, 12:42 AM
Reason: asfdasdf
Reason: asfdasdf
Furthermore, the 
relation is the same as 
by angle bisector theorem, which we can simplify to 
using the previous equation. Thus, it suffices to prove this last relation.
implies 
and 
,
.
lies on the rectangular hyperbola with foci 
.
homothety at 
,
,
,
.
be the harmonic conjugate of 
.
Thus, 
.
so 
or 
,
lies on 
.
be the midpoint of 
is the Newton line of 
.
lies on 
,
satisfying 
.
.
,
,
be the reflection of 
with 
,
