be real numbers, such that![\[
a^2+b^2+c^2+abc=5
\]](http://latex.artofproblemsolving.com/7/c/8/7c8044347b7e25cb21dbef42c65ec351cefd4699.png)
![\[
a+b+c=3
\]](http://latex.artofproblemsolving.com/6/4/a/64a3034ae02dd28aa6220ee5c37a22d608074173.png)
Prove that atleast one of the numbers 
.
.
?
?
(i.e., the square on 
has 
,
,
to the centers of the opposite squares are concurrent.
be the solid square region with vertices at![\[
(0, 0), \left(\frac{1}{2}, \frac{1}{2}\right), (1, 0), \left(\frac{1}{2}, -\frac{1}{2}\right).
\]](http://latex.artofproblemsolving.com/a/9/8/a9898f9237f07eaef5d17a9ac689d80ca631863f.png)
consists of two copies of 
:
counterclockwise around the origin and scaled by a factor of 
,
.
below.
be the limiting region of the sequence 
.
for relatively prime positive integers 
and 
.
.
,
.
.
be prime numbers such that![\[
\frac{1}{pq}+\frac{1}{qr}+\frac{1}{pr}=\frac{1}{839}
\]](http://latex.artofproblemsolving.com/b/6/f/b6f882f3c93b0877419db625c2a427fc9561605b.png)
Hence find the numbers of ordered triples 
,
equal to 
,
equal to 
.
, where 
.
for all real numbers 
satisfying the equation 
is:
Y by
Let 
be a circumscribed quadrilateral with incircle 
and no two opposite angles equal. Let 
be an arbitrary point on the diagonal 
, which is inside . The segments 
and 
intersect at 
and 
. The tangents to at and intersect at 
. Prove that lies on the line .
be a circumscribed quadrilateral with incircle 
and no two opposite angles equal. Let 
be an arbitrary point on the diagonal 
, which is inside . The segments 
and 
intersect at 
and 
. The tangents to at and intersect at 
. Prove that lies on the line .This post has been edited 1 time. Last edited by Assassino9931, Apr 9, 2025, 10:36 PM
at 
and let 
.
is the pole of 
and 
is the pole 
,
is the pole of 
and 
are collinear. Let 
and 
;
is harmonic, and since 
touch 
.
by projecting 
through 
,
.
such that the line 
passes through 
concur on 
,
,
concur, as desired.
