[/quote]
,
of relatively prime integers 
and 
such that 
is divisible by 
.
![\[\frac{a}{b^3+4}+\frac{b}{c^3+4}+\frac{c}{d^3+4}+\frac{d}{a^3+4}\]](http://latex.artofproblemsolving.com/4/d/2/4d21982ba412c0bb4a7985798e81bc5b01ac0e43.png)
given that 
,
,
,
are nonnegative real numbers such that 
.
be the set of all integers. Find all pairs of integers 
for which there exist functions 
and 
satisfying![\[ f(g(x))=x+a \quad\text{and}\quad g(f(x))=x+b \]](http://latex.artofproblemsolving.com/3/2/8/328146c3331a66cdc5cfc4a01b0609e91d278ec5.png)
for all integers 
.
satisfying 
.
such that 
be the sequence defined by 
and 
for each integer 
.
is a prime and 
is a positive integer. Prove that some term of the sequence 
.
is interesting if 2018 divides 
(the number of positive divisors of 
be integers and 
be a prime for which![\[ x^2-3xy+p^2y^2=12p \]](http://latex.artofproblemsolving.com/2/4/c/24cb51eb55144a9a1e0e489d115993496a052e79.png)
.
satisfying 
for all positive integers 
be an isosceles trapezoid with 
.
be the midpoint of 
.
and 
the circumcircles of the triangles 
and 
,
be the crossing point of the tangent to 
with the tangent to 
.
is tangent to 
,
denote the feet of the altitudes from 
,
.
and 
be the circles through 
,
other than 
such that![\[ f(x+f(y))=y+f(x+1),\]](http://latex.artofproblemsolving.com/b/a/3/ba35828951778c0d35f50b5b557796bd2cf9e338.png)
for all 
.
is defined by 
,
for 
.
of ordered pairs 
for which there is a finite set 
of positive integers such that 
,
.
,
,
with the following property: An ordered pair of nonnegative integers ![\[m < \alpha x+\beta y < M\]](http://latex.artofproblemsolving.com/a/9/b/a9bb35658173fdab821cc8180420e47d5437e0a2.png)
if and only if 
.
.
,![\[
f(xy)(f(x) - f(y)) = (x-y)f(x)f(y)
\]](http://latex.artofproblemsolving.com/0/9/b/09b8ed4e133ccf33b3f902fd9a9aa8e23c264341.png)
for all 
.
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
?
,
.
?
be a regular pentagon. Segments 
and 
intersect at 
?
.
,
?
,
and 
?
,
,
?
.
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 
on the curve 
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 
.
.
