be a scalene triangle with incircle 
.
the midpoint of arc 
in the circumcircle of 
the point where the 
-
.
meets 
and intersects 
,
.
or 
is tangent to 
,
,
are
with 
,
are all coloured blue.
for which the value of the expression![\[x^2+6x+33\]](http://latex.artofproblemsolving.com/3/d/e/3de5bc2a9b022192315975731d48af7fda9fe228.png)
is a perfect square.
for which there exist infinitely many positive integers 
such that ![\[ \frac{a^m+a-1}{a^n+a^2-1} \]](http://latex.artofproblemsolving.com/2/8/a/28a68e409451bbdcaee7c1286341f6aeea9ffdad.png)
is itself an integer.
Two circles 
and 
with radii 
and 
respectively are externally tangent. The straight line 
is tangent to the circles 
and 
respectively.The parallel line 
to the line 
and 
The line 
through 
and the circle 
and 
respectively, different from 
Y by
A line through the vertex 
of the triangle 
which doesn't coincide with 
or 
intersectes the altitudes from 
and 
at 
and 
respectively. Let 
be the reflection of in and 
be the reflection of in 
Prove that the circles 
and 
are tangent.
of the triangle 
which doesn't coincide with 
or 
intersectes the altitudes from 
and 
at 
and 
respectively. Let 
be the reflection of in and 
be the reflection of in 
Prove that the circles 
and 
are tangent.
and 
.
.
is cyclic. We can also see that 
,
,
have common tangent.
be foot of perpendicular line of C. Then 
done.
