be a triangle with incenter 
,
and 
respectively such that 
is a parallelogram. Points 
and 
are the angle bisectors of angles 
and 
respectively. Let 
be the circle tangent to segment 
past 
past 
decide whether there exist pairwise distinct positive integers 
such that for every 
,
divides 
.
and 
.
such that ![\[f(xy-1)+f(x)f(y)=2xy-1,\]](http://latex.artofproblemsolving.com/8/8/8/888ca39f2b7f8cec6d6426bee28d40eade40a66e.png)
for any real numbers 
and 
,
fractions 
The product of these fractions equals 
,
be a function satisfying
for all integers 
and 
.
and 
for all integers 
such that: 
.
of the triangle 
which doesn't coincide with 
or 
intersectes the altitudes from 
and 
at 
and 
respectively. Let 
be the reflection of 
be the reflection of 
Prove that the circles 
and 
are tangent.
be positive real numbers such that 
Prove that we have![\[(a+b)(b+c)(c+a)\geq 8.\]](http://latex.artofproblemsolving.com/2/a/6/2a63f19de634fce70ff4ebc02ad13d442a20c378.png)
such that for any positive integer 
there exists a positive integer 
are equal to 
such that for any positive real numbers 
,![\[
\frac{xy}{x^2 + y^2 + xz} + \frac{yz}{y^2 + z^2 + yx} + \frac{zx}{z^2 + x^2 + zy} \geq C.
\]](http://latex.artofproblemsolving.com/9/1/c/91c8a4af2bd2f3107c8921d2e06f8a09dedd0164.png)
Y by
Let the incircle of 
touch sides 
, 
, and 
at points 
, 
, and 
, respectively. Let 
be the diametrically opposite point of with respect to the incircle. Let lines 
and 
intersect the incircle again at 
and 
, respectively. Prove that the lines 
, 
, and 
are concurrent, i.e., the lines intersect at the same point.
(Kritesh Dhakal, Nepal)
touch sides 
, 
, and 
at points 
, 
, and 
, respectively. Let 
be the diametrically opposite point of with respect to the incircle. Let lines 
and 
intersect the incircle again at 
and 
, respectively. Prove that the lines 
, 
, and 
are concurrent, i.e., the lines intersect at the same point.(Kritesh Dhakal, Nepal)
This post has been edited 2 times. Last edited by jenishmalla, Mar 15, 2025, 3:00 PM
Reason: formatting
Reason: formatting
and take a homography preserving the circle and taking 
to the centre of the circle, from where it's obvious.
.
and let 
.
Since 
,
lies in a circle with diameter 
.
is a cyclic pentagon.
,
.
,
and 
is a harmonic quadrilateral. Brianchon's Theorem on hexagon 
,
and 
concur on line 
.
,
be the center of spiral similarity that sends 
that 
as 
meet the incircle again at 
to be the unit circle.
.
are collinear 
.
are collinear 
.
.
,
and we're done.
say 
is pole of 
(and 
is pole of 
)
lies on 
by La Hire. Now 
,
,
.
finishes.
![\[
P = EF \cap AD, \quad Q = EF \cap AD'.
\]](http://latex.artofproblemsolving.com/4/c/4/4c43ce3efdb65c792f4ba9800e4cc4b4742eef74.png)
![\[
(A, Q; D', X) = (A, P; Y, D) = -1.
\]](http://latex.artofproblemsolving.com/1/4/2/142bb90c50b44b6dfe0c36d97d43bdfe10f3b5cc.png)
and 
are defined by the intersections of 
and 
are harmonic divisions.
and 
and if 
