[/quote]
,
be a convex quadrilateral. Suppose that the circles with diameters 
and 
intersect at points 
and 
.
and 
.
,
,
attaining all positive values and positive numbers 
are given. Numbers 
satisfy
for every 
.
satisfying 
.
?
be reals such that 
and 
.
and 
points on a plane is 
,
such that: in every set of 
points of a plane, no three of which are collinear, consisting of 
consisting of 
disting positive integers smaller than 
are given. Prove that there exists a positive integer 
that can be written in the form 
,
in at least 
different ways.
,
be the set of all positive integers not exceeding ![\[
P_n(x) = \sum_{k \in S_n} x^{k - 1}
\]](http://latex.artofproblemsolving.com/3/a/c/3ac60cb70baf4dfcf297e9af87480a8ee34ba70c.png)
and a polynomial 
with integer coefficients such that![\[
P_n(x) = (x^{r_n} + 1) Q_n(x).
\]](http://latex.artofproblemsolving.com/7/0/e/70e19a9d3e0a7e97abc7aa08d3f23126c14ea7d0.png)
is irreducible in ![$\mathbb{Z}[x]$](http://latex.artofproblemsolving.com/4/b/7/4b7d768fffbb7a1a6163af392ed348f2d49d8783.png)
.
integers are such that 
is prime and 
.
such that 
.
Y by RadioActive, nosaj, MathLearner01, flamefoxx99, football017, Benq, mathsolver101, chessderek, kj2002, greenchicken, C-bass, Adventure10, Mango247, and 9 other users
I recently (as in in the shower yesterday) had the idea of making a mock MATHCOUNTS competition. Unlike most mock competitions, I was thinking of releasing multiple tests (School, Chapter, State, and possibly National Mock Tests) over the course of the whole mock Mathocunts "season".
The competition will go as follows: I will release each level of the competition every month or so. You will have a 1-2 week window to officially submit your test answers, through PM. For the PM, please refrain yourself to use LaTex! Write fractions in the form A/B,
as pi, and square roots in the form sqrtx. Every test will have a raw maximum of 46, and the raw score is calculated by 
. Taking the raw scores, your official score will be calculated by ![$2\cdot\text{[School Raw Score]} + 3\cdot\text{[Chapter Raw Score]} + 4\cdot\text{[State Raw Score]} + 5\cdot\text{[National Raw Score]} = \text{Final Score}$](//latex.artofproblemsolving.com/8/9/7/8973b84c52be08db279270a1cd0f74fdb1505dac.png)
. After doing some math, you will find that the maximum score is a score of 
. If you submit your answers after the window is over, I will only grade it and send it back to you, deeming it useless contributing to your mock season score.
Each test packet contains a traditional 30 questions for sprint, and a traditional 8 questions for the Target Round. Every question in the sprint is designed such that calculators are unnecessary in solving the problem, so calculators are strcitly prhibited for the Sprint Round of the portion, with a time limit of 40 minutes. On the contrary, the Target Round is designed to have the assistance of a non-computer calculator, so follow accordingly. Instructions are thoroughly explained in each test packet. Feel free to print out the packet to solve or keep!
When submitting your answers, submit them to Th3Numb3rThr33 (dur). Do NOT include labels (such as "1337 Leets"), and please number your answers. While breaking the above two recommendations will not result in a penalty, it will make it extremely easy for me to grade the tests (using diffchecker.com).
By the way, this was inspired by the Mock AMC 6, and iNomOnCountdown's Mock MATHCOUNTS test. Check them both out!
=======================================================================
EDIT: The Chapter Competition is up and running! Deadline for submissions is February 28, 2015, at 11:59 PM. Do the test while you can!
ERRATA LIST FOR CHAPTER COMPETITION:
The competition will go as follows: I will release each level of the competition every month or so. You will have a 1-2 week window to officially submit your test answers, through PM. For the PM, please refrain yourself to use LaTex! Write fractions in the form A/B,
as pi, and square roots in the form sqrtx. Every test will have a raw maximum of 46, and the raw score is calculated by 
. Taking the raw scores, your official score will be calculated by ![$2\cdot\text{[School Raw Score]} + 3\cdot\text{[Chapter Raw Score]} + 4\cdot\text{[State Raw Score]} + 5\cdot\text{[National Raw Score]} = \text{Final Score}$](http://latex.artofproblemsolving.com/8/9/7/8973b84c52be08db279270a1cd0f74fdb1505dac.png)
. After doing some math, you will find that the maximum score is a score of 
. If you submit your answers after the window is over, I will only grade it and send it back to you, deeming it useless contributing to your mock season score.Each test packet contains a traditional 30 questions for sprint, and a traditional 8 questions for the Target Round. Every question in the sprint is designed such that calculators are unnecessary in solving the problem, so calculators are strcitly prhibited for the Sprint Round of the portion, with a time limit of 40 minutes. On the contrary, the Target Round is designed to have the assistance of a non-computer calculator, so follow accordingly. Instructions are thoroughly explained in each test packet. Feel free to print out the packet to solve or keep!
When submitting your answers, submit them to Th3Numb3rThr33 (dur). Do NOT include labels (such as "1337 Leets"), and please number your answers. While breaking the above two recommendations will not result in a penalty, it will make it extremely easy for me to grade the tests (using diffchecker.com).
By the way, this was inspired by the Mock AMC 6, and iNomOnCountdown's Mock MATHCOUNTS test. Check them both out!
=======================================================================
EDIT: The Chapter Competition is up and running! Deadline for submissions is February 28, 2015, at 11:59 PM. Do the test while you can!
ERRATA LIST FOR CHAPTER COMPETITION:
This post has been edited 15 times. Last edited by Th3Numb3rThr33, Feb 21, 2015, 11:54 PM
