[/quote]
,
be positive real numbers such that 
.
-
,
is a digit. How many pairs of such numbers are there such that their sum is also a palindrome?
,
.
,
and 
.
is prime, 
with 
and 
w![$ \boxed{\ \left(\frac{n}{n+1}\right)^{n+1}<\sqrt [n+1]{(n+1) !}-\sqrt [n]{n!}\ }$](http://latex.artofproblemsolving.com/d/f/2/df2545bd7a1f1759e55190e60648d5cf65a2488a.png)
be the incircle of 
.
amd 
inside 
,
are concyclic.
such that ![\[p^q+q^p+p+q-5pq\]](http://latex.artofproblemsolving.com/b/a/b/bab8a0ee3555cf1136b6de56179012b920d0a2ad.png)
is a perfect square.
be positive numbers such that 
.![\[T = \dfrac{bc}{\sqrt{a}+3}+\dfrac{ca}{\sqrt{b}+3}+\dfrac{ab}{\sqrt{c}+3} \]](http://latex.artofproblemsolving.com/6/d/5/6d570d06679f17452790b90171428d645e97fe9f.png)
with circumcircle 
,
on 
be points on lines 
,
,
,
concur at a point 
be a point on line 
,
,
be a point on line 
be a point on line 
,
,
concur at a single point.
be non-negative real numbers such that 
Prove that
Prove that 
.
.
Y by
In a convex quadrilateral 
, diagonals 
and 
are equal, and they intersect at 
. Perpendicular bisectors of 
and 
intersect at point 
lying inside triangle 
, and perpendicular bisectors of 
and 
intersect at point 
lying inside triangle 
. Prove that 
.
, diagonals 
and 
are equal, and they intersect at 
. Perpendicular bisectors of 
and 
intersect at point 
lying inside triangle 
, and perpendicular bisectors of 
and 
intersect at point 
lying inside triangle 
. Prove that 
.
