[/quote]
,
on a plane with centers 
,
is feet of the perpendicular from 
to 
and 
and 
are intersections of line 
with circle 
.
and 
can move around the circle 
and 
,
.
prove that
is a prime number. We have a number of cards, each of which has a number written on it such that each of the numbers 
appears at most once and 
exactly once. To design a game, for each pair of cards 
and 
,
If card 
,
If card 
,
must not win over card 
.
of natural numbers such that for every pair of natural numbers 
and 
,![\[
\frac{1}{2} < \frac{\gcd(a_r, a_s)}{\gcd(r, s)} < 2
\]](http://latex.artofproblemsolving.com/1/6/7/167679c1707b957d87311298ea5b72347a9bdc45.png)
such that
the following equation :
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 triangle with 
.
be a point on 
such that 
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 
.
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 
.
.
