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.
has three distinct real roots 
, 
and 
. The polynomial 
has roots ![$\sqrt[3]{\alpha}$](http://latex.artofproblemsolving.com/4/f/5/4f57cc54c79420c509385c423ffa42820cef3109.png)
, ![$\sqrt[3]{\beta}$](http://latex.artofproblemsolving.com/d/4/0/d40f4039c320468d9974a9504917b56eaf73c947.png)
, ![$\sqrt[3]{\gamma}$](http://latex.artofproblemsolving.com/7/8/1/78171ba6cd442d8156fd70b3d845075e3d82a926.png)
. Find the integer closest to 
.
is a convex quadrilateral with![\[\angle BAC = 30^\circ \]](http://latex.artofproblemsolving.com/c/c/b/ccb5d05e9588fe1fa56d66e7cdb8b4ed5b68dcc2.png)
![\[\angle CAD = 20^\circ\]](http://latex.artofproblemsolving.com/e/2/e/e2e626ead23ee348b788daccaeba6fcda6d48831.png)
![\[\angle ABD = 50^\circ\]](http://latex.artofproblemsolving.com/e/c/8/ec8edd2c2ee961f0ea87bd5cc67924fe26976b90.png)
![\[\angle DBC = 30^\circ\]](http://latex.artofproblemsolving.com/5/5/e/55e034d2157a60bf72ba777b59ad893b51cc1306.png)
, show that 
.
be a triangle and 
its centroid. A point is taken on segment 
. Take points 
and 
on the sides 
and 
respectively, so that 
and 
. Show that as varies in segment , the circumcircle of triangle 
passes through a fixed point 
.
be positive integers.Prove that there are no positive integers on the interval 
.
is an integer greater than 1, let 
be the greatest prime factor of 
(
and i is an integer).
greater than 1, there exists a natural number
, such that 
be the number of different numbers in 
. Find the largest number in 
and give reasons. be the number of different numbers in . Find the largest number in 
and give reasons.
verifying :
for all 
is inscribed in a circle of radius 
, such that the diagonal is a diameter and 
. Diagonals intersect at . It is known that 
. What is the length of side 
?
with the following property: for all positive divisors 
of , we have 
or 
is prime.
be an acute triangle with orthocenter 
and circumcircle 
. Let 
be the midpoint of side . Point 
is chosen from the minor arc on 
such that 
. Let 
be a point on such that 
is perpendicular to 
, and 
be a point on line such that 
is perpendicular to . Lines 
and intersect at point 
, and point is the reflection point of with respect to .
.
for all 
.
and 
Prove that
such that
x>y>0
2
