AoPS Academy
be an acute triangle with 
and let 
be its orthocenter. Let 
be a point on the perpendicular bisector of 
such that 
and and 
are on different sides of 
, 
a point on the perpendicular bisector of 
such that 
and 
a point on the perpendicular bisector of 
such that 
and 
lie on the opposite side of 
w.r.t 
. Prove that 
and are collinear.
be a polynomial with integer coefficients such that 
, and let 
be an integer. Define 
and 
for all integers 
. Show that there are infinitely many positive integers 
such that 
.
be the roots of the polynomial 
. Find 
if within that number exists a digit 
that appears exactly 
times in that number, hence find the number of that consist of 4 digits (Numbers may contain a 0)
be the real roots of![\[x^3 + 7x^2 -6x - 13 = 0\]](http://latex.artofproblemsolving.com/6/6/3/6639be4c9e92b5ad4466359681db0d2a461ec8dd.png)
Find 
, 
, and 
be the roots of 
Find the numerical value of the expression:![\[\frac{\alpha + 2}{\alpha - 2} + \frac{\beta + 2}{\beta - 2} + \frac{\gamma + 2}{\gamma - 2}\]](http://latex.artofproblemsolving.com/b/5/9/b59f4515409422333e10857ea775a36e58490b5d.png)
and 
such that 
and 
. Find the value of 
on one variable with real coefficients such that the equation 
has exactly 
ordered pairs 
as solutions?
such that 
coprime, 
and 
.
and 
such that:
be distinct non-zero real numbers and consider the polynomial 
Since 
for all real numbers, 
.
.
and at lest one of 
and 
must lie in 
.
be the number of ordered integer triplets 
,
,
.
.
.
.
.
and 
giving