[/quote]
,
to 
have the digit 
and are divisible by 
?
be a polynomial with integer coefficients. Suppose there exist infinitely many integer pairs 
such that 
.
![\[\sqrt x - \frac 1y - 2w + 3z = 1,\]](http://latex.artofproblemsolving.com/d/0/f/d0ff4819db851e6c12f738dcdb6b81bbd2ed0ca7.png)
![\[x + \frac{1}{y^2} - 4w^2 - 9z^2 = 3,\]](http://latex.artofproblemsolving.com/6/5/3/653c1547620f85ff761817a5cc333f16bce9c4c6.png)
![\[x \sqrt x - \frac{1}{y^3} - 8w^3 + 27z^3 = -5,\]](http://latex.artofproblemsolving.com/1/b/9/1b9c2ed383acc931aa3607ba82031e182e18f3d1.png)
![\[x^2 + \frac{1}{y^4} - 16w^4 - 81z^4 = 15.\]](http://latex.artofproblemsolving.com/e/0/3/e032ce109cd74bbae47dc6bca04dcab2d8f9ef29.png)
![\[
|MA|^2 + |MB|^2 + |MC|^2 = |GA|^2 + |GB|^2 + |GC|^2 + 3|MG|^2
\]](http://latex.artofproblemsolving.com/0/7/0/070ff15920d19d474defb77ceb606b2811fa85b4.png)
such that the numbers ![\[ (a_1)^{2018}+a_2, (a_2)^{2018}+a_3, \dots, (a_{2018})^{2018}+a_1 \]](http://latex.artofproblemsolving.com/6/1/a/61ad92b49bd9b4b8839e6f6c4e81c758a11835a3.png)
are all powers of 
-
?
students want to share 
pencils. If every student gets at least one pencil, how many ways are there to distribute the pencils?
?
of positive integers satisfying 
and 
for all positive integers 
and 
For a given positive integer 
find the maximum possible value of 
such for any 
:
.
and 
be nonzero real numbers satisfying![\[
a_1^2 b_1^2 (a_1 + b_1) + a_2^2 b_2^2 (a_2 + b_2) + \cdots + a_n^2 b_n^2 (a_n + b_n) \leq 7,
\]](http://latex.artofproblemsolving.com/a/7/e/a7ea7af24ac529090f3b2e305be43e2728b79eb4.png)
![\[
\frac{1}{a_1} + \cdots + \frac{1}{a_n} = \frac{1}{4}, \quad \frac{1}{b_1} + \cdots + \frac{1}{b_n} = \frac{1}{3}.
\]](http://latex.artofproblemsolving.com/e/f/3/ef341b850979a9b780f5daaa49f46f1d1324cdbd.png)
Find the maximum value of![\[
a_1 b_1 + a_2 b_2 + \cdots + a_n b_n.
\]](http://latex.artofproblemsolving.com/d/3/3/d338bf5be4f1a218e150bfd84173e988ee92ddad.png)
![\[
P(x) = a_0 + a_1 x^2 + a_2 x^4 + \cdots + a_{10} x^{20}
\]](http://latex.artofproblemsolving.com/0/d/c/0dcd08456f253de08f3b880c7a1fddc07b335932.png)
be a polynomial of degree 20 with only even powers of 
.
be 
.![\[
(x_1^2 + 1)(x_2^2 + 1) \cdots (x_{20}^2 + 1) = 2025,
\]](http://latex.artofproblemsolving.com/b/a/4/ba4bc5f526565e56a9a685fe01572c1366241084.png)
find the **minimum value** of 
.
Y by CaptainFlint, DeathLlama9, Mudkipswims42, uwotm8, dantx5, Benq, pandyhu2001, PlatinumFalcon, 24iam24, suvamkonar, Adventure10, Mango247
Hello everyone! Many AoPSers have been collaborating over the past month to write a mock MATHCOUNTS competition. This competition will be around State/National level.
Update: The test has been released! It is attached to this post. The deadline will be May 31, 2015.
How to submit answers: Send a private message to me (nosaj) with your answers. Separate each of your answers with a &. To leave a question blank, submit 0 as the answer. For example, if you thought that the answers to the Sprint Round were all 42 and the answers to Target 2-7 were all 1337, but you left Target 1 blank, you would submit
See the third page of the PDF for answer formatting directions. Do not use LaTeX!
The submission period is over now. See this post for the answers and results. Thanks to everyone who participated!
This post has been edited 7 times. Last edited by nosaj, Jun 4, 2015, 4:08 AM
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