AoPS Academy
be a cubic polynomial given by 
, where 
are integers and 
. Suppose that 
for infinitely many pairs 
of integers with 
. Prove that the equation 
has an integer root.
be real numbers such that 
. Determine the minimum value of 
and determine all values of 
such that the minimum value is achived.
cities. The airline Starways wants to establish some one-way flights between pairs of cities in such a way that each city has exactly one flight out of it. Find the smallest positive integer 
such that no matter how Starways establishes its flights, the cities can always be partitioned into groups so that from any city it is not possible to reach another city in the same group by using at most 
flights.
distinct circles in a plane, with radius 
.
circles, which form a set 
, which satisfy the following., they intersect with each other or, they don't intersect with each other.
be a triangle, and let 
be on 
so that 
. Let 
intersect the circumcircle of 
again at 
. Let 
be the midpoint of 
and let the perpendicular bisector of 
intersect 
at 
. If 
is the circumcenter of 
, prove that 
.
be a triangle, and let be on so that . Let intersect the circumcircle of again at . Let be the midpoint of and let the perpendicular bisector of intersect at . If is the circumcenter of , prove that .
and 
is perpendicular to 
and hence we just need to prove that lies on their radical axis. Power of to second circle is just 
and see that 
and hence 
and so power of to first circle is 
and obviously 
, done.
not containing 
.
,![\[\angle ZAD = \angle ZAM = \angle AMZ = \angle AMC = \angle ABC = \angle ABD\implies AZ \text{ is tangent to } \odot(ABD).\]](http://latex.artofproblemsolving.com/2/6/1/2617bd6fc6402446013b1b624b242db161020119.png)
Also![\[\angle MBD = \angle MBC = \angle MAC = \angle ABD \implies BM \text{ is tangent to } \odot(ABD).\]](http://latex.artofproblemsolving.com/a/c/b/acb407e74dc3f6c5b7b204b11193c667fef865f3.png)
Now 
is tangent to 
and 
lies on the radical axis of 
.
is tangent to 
lies on the radical axis of 
.
and 
are tangent to 
.
,
.
and 
.
is the radical axis of 
.
be the reflection of 
perpendicular to 
at 
at 
.
,
.
,
.
,
lie on circle 
.
by 
.
.
This means that 
,
must be the radical axis of 
is the line joining the centers of these two circles, which is perpendicular to the radical axis 
respectively, so 
.
.
tangent to 
be a circle centred at 
.
is tangent to 
.
is tangent to 
is the radical axis. Done