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.
. Prove that if the leading coefficient of the polynomial equals then the greatest common divisor of the other coefficients also equals .
be a triangle with circumcenter 
. Point 
is the intersection of the parallel line from to 
with the perpendicular line to 
from 
. Let 
be the point where the external bisector of 
intersects with . Let 
be the projection of onto 
. Prove that the lines 
have a common point.
dwarves, whose weigths are 
, who want to cross a river. They have a small boat which can lift at most 
each time without sinking. For each journey of the boat a non-empty subset of the dwarves to be taken to the other side is chosen and one of these dwarves is chosen as the 
for that journey. Since return journeys are counter-current, no dwarf is able to do the rower for more than one return journey. Is it possible for all the dwarves to reach the other side of the river?
with circumcenter and orthocenter 
, let 
be an arbitrary point on the circumcircle of triangle such that does not lie on line 
and that line 
is not parallel to line 
. Let 
be the point on the circumcircle of triangle such that 
is perpendicular to , and let 
be the point on line such that 
. Let 
and 
be the points on the circumcircle of triangle such that 
is a diameter, and 
and 
are parallel. Let 
be the midpoint of 
.
and 
are perpendicular.
and 
are perpendicular.
with 
, a tangent to the circumcircle of at point 
intersects the extension of 
at point 
. 
is the midpoint of 
, and 
intersects the circumcircle of at 
. Prove that 
.
be positive real numbers such that 
. Prove that![\[ \frac{a_{1} a_{2} \cdots a_{n} \left[ 1 - (a_{1} + a_{2} + \cdots + a_{n}) \right] }{(a_{1} + a_{2} + \cdots + a_{n})( 1 - a_{1})(1 - a_{2}) \cdots (1 - a_{n})} \leq \frac{1}{ n^{n+1}}. \]](http://latex.artofproblemsolving.com/6/5/5/65558feba82e32266d3d3cbdfea85e079483403f.png)
with integer coefficients such that for all positive number 
and prime 
satisfying 
, we have 
.
, where 
, then the number of integers in the range of 
is __
be a triangle with 
. Let the angle bisector of 
intersects at and intersects the perpendicular bisector of segment at 
. Prove that 
lies on the circle around 
.
Of course the quad 
is cyclic, thus Ptolemy's theorem holds, and 
can be expressed as:
Now onto some trigonomtry:
.
and on 
.
):
Now we have the following when we input this into our ratio (
):
Thus we need to show that 
and representing 
.This statement get's boiled down to the following:
Which is an identity, thus the statement of the problem is true..... 
lies on the circumcircle of . By Ptolemy's theorem on 
and since 
, ![\[AQ\cdot BC=(AB+AC)QB\;\Longrightarrow\;\frac{QB}{AQ}=\frac{BC}{AB+AC}.\]](http://latex.artofproblemsolving.com/0/5/2/052ade653cce1a7aed44d3c0f27ca76e17f38005.png)
by Shooting lemma (or 
): we have ![\[\left(\frac{BC}{AB+AC}\right)^2=\frac{QA\cdot QP}{AQ^2}=\frac{PQ}{AQ}.\]](http://latex.artofproblemsolving.com/b/8/8/b881bae336ee6d38271bd5bff58a97dc4b5bc78d.png)
since is cyclic
, which can easy obtain through a 
inversion 
prove that![$\frac{a}{b} + \sqrt {\frac{b}{c}} + \sqrt[3]{{\frac{c}{a}}} > \frac{5}{2}$](http://latex.artofproblemsolving.com/6/a/6/6a657cb647e279b0745e3189c3f9d54aa25bf976.png)
be reals such that 
P
satisfying 
.
.
,
and 
.
and 
so 
.
.