[/quote]
,
be nonnegative real numbers such that 
.
and 
of the same colour such that 
is the midpoint of 
.
a triangle with circumcircle 
and isogonal conjugates 
and 
.
and 
points on 
and 
respectively such that 
.
cut 
.
and 
cut at 
chess board. Every link of the chain connects the centers of adjacent squares either horizontally, vertically or diagonally, where the two squares are adjacent if they share an edge or a corner. For the interior polygon bounded by the chain, prove that the total area of black pieces equals the total area of white pieces. “
,
,
cut three common chords concurrent at 
,
which are internally tangent to all 
.
.
be a point on 
,
be a line passing through 
.
at 
and the circumcircle of triangle 
at 
)
and 
are tangent at 
be a set of integers with the following properties:
.
and 
,
.
,
is composite, then all positive divisors of 
such that 
is an integer for every positive integer 
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
?
and 
,
.
such that 
?
be a regular pentagon. Segments 
and 
intersect at 
.
?
.
,
?
be a square. Let 
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 
.
.
.
,
?
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 
.
.
