AoPS Academy
[/quote]
, do not answer with "
is a solution" only. Either you post any kind of proof or at least something unexpected (like "
is the smallest solution). Someone that does not see that is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
are the roots of the polynomial 
. Let 
be the unique monic polynomial whose roots are 
. Find 
.
are in arithmetic progression. Find the value of 
.
, 
, 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 
be arbitrary points on the sides 
and 
of triangle 
, respectively. The circumcircle of 
meets for the second time the circumcircle of 
at point 
. Line 
meets line 
at point 
, while line 
meets line 
at point 
. Prove that lines 
and 
pass through the same point.
with circumcenter 
. 
and 
intersect the altitude from 
to at points 
and 
respectively. 
is the circumcenter of triangle 
and 
is the reflection of over . 
is the second intersection of circumcircles of triangles 
and 
. Show that 
are collinear.
is the center of the inscribed circle and holds 
. Prove that 
.
be an acute triangle with incentre 
and 
. Let lines 
and 
intersect the circumcircle of at 
and 
, respectively. Consider points 
and 
such that 
and 
are parallelograms (with 
, and 
). Let 
be the point of intersection of lines 
and 
. Prove that points 
, and are concyclic.
, 
, 
, and is the point such that 
and 
, where and 
lie on the same side of 
. Let be the point on such that 
, and let 
be the midpoint of 
. Prove that the line through parallel to 
passes through the midpoint of .
, points and lie on and respectively such that 
. Points 
and are the reflections of and with respect to the midpoints of sides and , respectively. Point lies on segment 
such that the circumcenter of triangle 
coincides with the circumcenter of triangle . passes through the midpoint of segment 
.
be a triangle with a circle 
with center 
tangent to sides 
at 
respectively. Suppose the circle with diameter 
intersects the circumcircle of again at 
is the reflection of over . Suppose points 
lie on such that 
are parallel to . Prove that: The intersection of 
lie on the circumcircle of 
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 triangle with points 
lie on the perpendicular bisector of such that 
lie on a circle. Suppose 
are perpendicular to sides 
at points 
The tangent lines from points to the circumcircle of 
intersects at point 
Prove that: 
are parallel.
, consider the generating function 
. Clearly, this yields 
. Therefore, we obtain
, we have
, it is straightforward to observe that the coefficient of 
in 
is 
. This leads to the conclusion.
, consider the generating function . Clearly, this yields . Therefore, we obtain, we have, it is straightforward to observe that the coefficient of in is . This leads to the conclusion.
and 
where each summation runs over 
variables 
and 
Then find the value of 
in 
is 
,
.
.
.
.
.
.
.
.
.