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.
be an integer greater than zero. Then, the value of the sum of the reciprocals of the cubes of the roots of the equation![\[
mx^4 + 8x^3 - 139x^2 - 18x + 9 = 0
\]](http://latex.artofproblemsolving.com/a/3/a/a3a9bca412e927632e4921c51f2a6593409031d9.png)
is equal to:
and 
be on segment 
of an acute triangle 
such that 
and 
. Let 
and 
be the points on 
and 
, respectively, such that is the midpoint of 
and is the midpoint of 
. Prove that the intersection of 
and 
is on the circumference of triangle .
, then prove that the following inequality holds:![\[
\sum_{\text{cyc}} \sqrt{1 + \left(x\sqrt{1 + y^2} + y\sqrt{1 + x^2}\right)^2} \geq \sum_{\text{cyc}} xy + 2\sum_{\text{cyc}} x
\]](http://latex.artofproblemsolving.com/9/5/5/955863a0cf8747ae45b736b0631f243d3908eb84.png)
be triangle, inscribed in parabola. Tangents in points 
forms triangle . Prove that 
. (
is area of triangle 
).
such that the equality 
holds for all pairs of real numbers 
.
, in which 
: 
, but the proof is important.
be an integer. Find all sequences 
satisfying the following conditions:![\[ \text{ (a) } a_i \in \left\{0,1\right\} \text{ for all } 1 \leq i \leq n^2 + n;
\]](http://latex.artofproblemsolving.com/3/c/5/3c509ec2e9013e8d3be492c8eb44a7c33841b74e.png)
![\[ \text{ (b) } a_{i + 1} + a_{i + 2} + \ldots + a_{i + n} < a_{i + n + 1} + a_{i + n + 2} + \ldots + a_{i + 2n} \text{ for all } 0 \leq i \leq n^2 - n.
\]](http://latex.artofproblemsolving.com/9/7/d/97d2a467d1c0dc8594ec024c3bb9b8c87ee85b19.png)
be an acute triangle with circumcircle 
. Let 
be the midpoint of 
and let 
be the midpoint of 
. Let 
be the foot of the altitude from 
and let 
be the centroid of the triangle . Let 
be a circle through and that is tangent to the circle at a point 
. Prove that the points 
and 
are collinear. with integer coefficients such that ![$P\left(\sqrt[3]5+\sqrt[3]{25}\right)=5+\sqrt[3]5$](http://latex.artofproblemsolving.com/5/8/e/58ec2a921cd988bd5ae017a8653544ddd758225f.png)
.
be a 
-element subset of the set 
. Prove that there exist numbers 
, 
in 
such that the sets ![\[ A_j=\{x+t_j\mid x\in A\},\qquad j=1,2,\ldots,100 \]](http://latex.artofproblemsolving.com/d/5/c/d5c2a7b943e8c7200361698f3eb6f83700ad68d6.png)
are pairwise disjoint.
positive integers 
such that for every index 
, with 
, 
is a multiple of 
. Determine the greatest possible value of the largest of the numbers.
be a triangle. Let and be the points in which the median and the angle bisector, respectively, at meet the side . Let and be the points in which the perpendicular at to 
meets 
and 
, respectively. And 
the point in which the perpendicular at to meets produced.
is perpendicular to .
be the Fermat point of a 
. Prove that the Euler line of the triangles 
, 
, 
are concurrent and the point of concurrence is , the centroid of .
until such that 
is equilateral now, taking as homothetic center the midpoint of (3:1) sending X to circumcenter of , P to centroid of and A to centroid of that's enough
, 
and 
are the bisectors of 
, 
cutting 
, at , 
respectively. If 
and 
, find the angles of 
, and are the bisectors of , cutting , at , respectively. If and , find the angles of
, be the midpoint of . Prove that
.
and 
are points on the circumcircle of such that 
is parallel to . Prove that 
is perpendicular to the Simson line of .
be the Fermat point of a . Prove that the Euler line of the triangles , , are concurrent and the point of concurrence is , the centroid of . then the problem is trivial by Liang-Zelich , consider the midpoint of , 
, as 
,let the centroid of be 
then we know 
so 
is parallel to 
which is also parallel to . now meets the point outside of , as 
such that 
is an equilateral triangle, and now by 
, where 
is the midpoint of we have 
is parallel to so 
lies on therefore it's done
be an acute triangle whose incircle touches sides and at and 
, respectively. Let the angle bisectors of 
and 
meet 
at and 
, respectively, and let the midpoint of be 
. Show that 
is equilateral if and only if 
.
are drawn perpendicular to the sides 
respectively of then prove that
Not been able to solve this even though I know I can do this.
is parallel to , show that tan 
tan 
.
but nvm
be the orthocenter of and let be the circumcenter
from and be and 
then just use 
, is the circumcentre and is the orthocentre. Then, prove that 
.
be the antipode and then 
is a rhombus
is parallel to , show that tan tan . but nvm be the orthocenter of and let be the circumcenter from and be and then just use 
is an obvious solution. Now, let 
.
or 
.
.
now they are the circumcenters of the equilateral triangle formed outside of 
,
,
is concurrent at 
.
,