[/quote]
,
be a positive integer, and let 
denote the sum of the positive divisors of 
.
smallest positive integer relatively prime to 
-smooth if all of its prime factors are 
.
denote the number of 
.
be the least quadratic non-residue mod 
.
is a non-negative arithmetic function and
is convergent for some 
,
,
such that any graph with 
vertices and at least 
and 
,
be vertices. Define a directed graph 
as follows: for any integer point 
,
,
.
,![\( G[k, l] \)](http://latex.artofproblemsolving.com/6/3/a/63a5df1124231155f3240a3e21e24d293d0e8db3.png)
denote the subgraph of ![\([k, l]\)](http://latex.artofproblemsolving.com/2/9/f/29f08f190960a99d062fde7f2128bcb635657f36.png)
.
such that for any integer 
,![\( G[r, r + \alpha - 1] \)](http://latex.artofproblemsolving.com/5/1/a/51afcceb73daae97f8fc61456bf043b73b2689d7.png)
of ![\( G[r, r + \alpha] \)](http://latex.artofproblemsolving.com/8/8/7/887e55f5636b842e7b64c9c03982b0374cdf1c9e.png)
(i.e., how many connected components and what each component is like).
of a graph be partitioned into 
,
.
.
.
-
.
where no more than one circle is disjoint.
be a median in an acute-angled triangle 
.
is chosen on the line through 
tangent to the circumcircle of 
so that 
.
and 
.
lies on the line 
.
be a cyclic quadrilateral for which 
.
and 
externally on the sides 
and 
respectively.
meets 
at 
,
meets 
at 
and 
denotes circumcenter of 
and 
meets again on 
.
.
that does not contain 
and let 
be the midpoint of 
be the 
-
be the reflection of 
are concyclic.
be a convex pentagon with 
and 
For each point 
The least possible value of 
can be expressed as 
where 
and 
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 aeryde.xin, rdj5933mile5, dft, Amir Hossein, Adventure10, and 4 other users
Hi all!
Contest season is coming up! To help with the preparation, I'm planning to host a mock AMC 12 soon. I would like to make it an online contest, so that you guys can submit answers and we can post scores, and so on. I'm putting the tentative dates as the weekend of Jan 21-22, during which sending in answers is allowed. Of course, problems will be available for practice after that as well.
Sounds good? Keep practicing, and stay posted!
UPDATE: Problems have been posted; please see below. You have until 11:59PM Pacific Time, Sunday 22nd, to PM me the answers.
As usual, 6 points for a correct answer, 1.5 points for not answering, and 0 points for a wrong answer.
Contest season is coming up! To help with the preparation, I'm planning to host a mock AMC 12 soon. I would like to make it an online contest, so that you guys can submit answers and we can post scores, and so on. I'm putting the tentative dates as the weekend of Jan 21-22, during which sending in answers is allowed. Of course, problems will be available for practice after that as well.
Sounds good? Keep practicing, and stay posted!
UPDATE: Problems have been posted; please see below. You have until 11:59PM Pacific Time, Sunday 22nd, to PM me the answers.
As usual, 6 points for a correct answer, 1.5 points for not answering, and 0 points for a wrong answer.
This post has been edited 4 times. Last edited by python123, Jan 19, 2012, 1:45 AM
?
.
such that 
?
and 
intersect at 
.
?
.
,
?
be the midpoint of side 
,
be the intersection of segments 
and 
?
,
are constants. If 
,
?
.
be a point on 
and 
,
.
?
.
and 
are the roots of the equation 
,
?
.
terms of it?
.
?
be equilateral triangles. 
,
is perpendicular to 
,
be integers such that 
,
for 
.
?
be the sum of the digits of 
.
?
is defined by 
,
for 
.
approach?
are there such that 
and 
is divisible by 
?
and 
.
.
be a point on segment 
.
,
?
is tangent to the circle 
.
,
?
,
.
be real numbers. Suppose that 
and 
.
?
.
.
?
and a point 
is in which of the following range?
.
![\[ \frac{ad+bc}{bd}+\frac{ab+cd}{ac}=\frac{13}{2}\]](http://latex.artofproblemsolving.com/3/4/0/340f67afdfa74d3a4db98cf8a4c6dff97e7fd6e2.png)
![\[ \frac{ad+bc}{cd}+\frac{bc+ad}{ab}=9\]](http://latex.artofproblemsolving.com/4/a/6/4a68194f779595605b60d6d308e310525c4304ea.png)
yields![\[\frac{(ad+bc)(ab+cd)}{bd\cdot ac}=0 \]](http://latex.artofproblemsolving.com/4/a/e/4aea0550b49b53238762b9f3c0e50d2ce7e1e18d.png)
and 
.![\[ x+y=\frac{13}{2}\]](http://latex.artofproblemsolving.com/2/9/8/2984863e0c29cccc29c1b962f5a507b9cdc94608.png)
![\[ xy=9\]](http://latex.artofproblemsolving.com/d/2/d/d2d4be24cb9072334f27f5db2e8dc2d25a0a351f.png)
.
,
.
,
and 
.
.
everywhere, all we need to do is translate 
.
must equal the slope of the line 
.
However, I thought this test was too easy (22 looks like a challenge problem from a school textbook) and contained too many problems from previous years.
)
.
which does what?
.
.
.
take on, because 
ordered pairs, giving us the answer 
.
,
mod 
.
.
