be a non-constant monic polynomial with integer coefficients and let 
be an infinite sequence. Prove that there are infinitely many primes, each of which divides at least one term of the sequence 
.
positive real numbers such that 
. Prove that :
be an integer, and 
can divide 
let 
be all distinct prime divisors of 
is an integer. ( where 
is defined as the number of positive integers 
that are relatively prime to 
be real numbers . Prove that
Where 
Y by Adventure10, Mango247
In an acute-angled triangle 
the interior bisector of angle 
meets 
at 
and meets the circumcircle of again at 
. From perpendiculars are drawn to 
and 
, with feet 
and 
respectively. Prove that the quadrilateral 
and the triangle have equal areas.(IMO Problem 2)
Proposed by Soviet Union.
the interior bisector of angle 
meets 
at 
and meets the circumcircle of again at 
. From perpendiculars are drawn to 
and 
, with feet 
and 
respectively. Prove that the quadrilateral 
and the triangle have equal areas.(IMO Problem 2)Proposed by Soviet Union.
and that 
we get 
,
.
.
which is obiously true.
. Therefore,
![$[AKNM]=\frac {AN\cdot KM}{2}\ \stackrel{(*)}{=}\ \frac {bc\cdot\sin A}{2}=[ABC]$](http://latex.artofproblemsolving.com/9/f/7/9f766bb61a570a687826baf883118456bd0928db.png)
.
consider a point 
so that 
.
denote its projections 
with the circumcircle of 
. Prove that ![$[AKNM]=[ABC]\cdot\sin \widehat {(AL,KM)}$](http://latex.artofproblemsolving.com/b/7/8/b78c4ae745a6419da181e948f761d4293d887fd2.png)
.
