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such that![\[(a^2 + b^2 + c^2)^2 = 3(a^3b + b^3c + c^3a) + 1. \]](http://latex.artofproblemsolving.com/9/e/4/9e400fd807a9535ede983b75dcc307d8b75a35b6.png)
+1 w
by adding every 
?
be a triangle and the points 
and 
on 
, 
and 
on 
, and 
and 
on 
be such that 
and 
. The intersections of 
with 
and 
are 
and 
respectively, and the intersections of 
with 
and 
are 
and 
, respectively. Prove that the lines 
and 
are concurrent.
+1 w
. Three points 
are on the sides 
respectively such that 
. If 
, then find the length of 
?
, 
is a perfect square. Find the sum of all possible values of .
. Prove that
Let 
. Prove that
Let 
. Prove that
Let 
. Prove that
such that 
and 
share a common complex root?
![\[17! = 355687ab8096000\]](http://latex.artofproblemsolving.com/a/6/d/a6d147d3866c4f1fe6dac9481e53d1354891388b.png)
for some digits 
and 
. Find the two-digit number 
that is missing above.
be the polynomial with leading coefficent 1 and rational coefficents, such that ![\[p\left(\sqrt{3 + \sqrt{3 + \sqrt{3 + \ldots}}}\right) = 0,\]](http://latex.artofproblemsolving.com/0/f/4/0f472be9f2cede0221e1b4d970d65de94dba01b9.png)
and with the least degree among all such polynomials. Find 
.
such that 
is a positive integral power of a prime positive integer."
for positive integral values of 
be the largest perfect square that divides . Find the number of positive integers less than 
such that 
is divisible by 
but not 
.
intersect non-tangentially at a point . Let 
and 
be the centers of circles 
and 
. Let and 
be the intersection of 
with and respectively, which is different from . Show that 
is a parallelogram.
intersect non-tangentially at a point . Let and be the centers of circles and . Let and be the intersection of with and respectively, which is different from . Show that is a parallelogram.
are different real numbers. Prove that:
,
.
