of 
touch 
at 
,
,
denote the line tangent to 
.
such that 
.
.
and 
.![\[\sum_{i=1}^n (a_i-a_{i+1})^3\leq \frac{1}{2\sqrt{3}} (\sum_{i=1}^n a_i )^3,\]](http://latex.artofproblemsolving.com/8/1/0/810f07e3394370ce06aec1e7025bcf5cc5433d60.png)
where 
.
and walks randomly. The probability of any point reaching all adjacent points is equal. Find the probability of the ant reaching 
without passing through 
with integer coefficients and integer zeros for which 
.
be the sequence defined by 
and 
for each integer 
.
is a prime and 
is a positive integer. Prove that some term of the sequence 
.
be the set of positive integers. A function 
satisfies the equation ![\[\underbrace{f(f(\ldots f}_{f(n)\text{ times}}(n)\ldots))=\frac{n^2}{f(f(n))}\]](http://latex.artofproblemsolving.com/2/e/7/2e73ff2f75b103e9d6a4004a42e4ed7f4ee64a67.png)
for all positive integers 
.
.
Y by Adventure10, Mango247
In a triangle 
, 
. 
is the altitude of the triangle. 
is on such that 
( is in the triangle). and 
cut the circumcircle of in 
and 
respectively. if tangent to circumcircle of at cuts 
at 
, prove that ,
and 
are collinear.(25 points)
the exam time was 4 hours and 30 minutes.
, 
. 
is the altitude of the triangle. 
is on such that 
( is in the triangle). and 
cut the circumcircle of in 
and 
respectively. if tangent to circumcircle of at cuts 
at 
, prove that ,
and 
are collinear.(25 points)the exam time was 4 hours and 30 minutes.
This post has been edited 1 time. Last edited by goodar2006, Aug 8, 2010, 11:12 AM
to 
the intersection point of 
and 
.
be the point where 
be the point where the tangent of the circumcircle of 
.
and 
we have![\[tan \angle AP_1O= \frac{OA}{OP_1}\]](http://latex.artofproblemsolving.com/7/1/d/71db19d42bba03763660c33ee11249a729cabf12.png)
![\[sin\angle OP_2M= \frac{OM}{OP_2}\]](http://latex.artofproblemsolving.com/e/a/7/ea780a1ca8febc310d392b98c7f329f7f6b7694b.png)
,
we are done. But it´s straightforward to check that 
and 
.
we are done....
and 
are equal, we conclude 
,
