[/quote]
,
be the remainder when![\[100x^{100} + 99x^{99} + 98x^{98} + \cdots + 3x^3 + 2x^2 + x\]](http://latex.artofproblemsolving.com/d/5/4/d5401c220a8682335a1306ae790a80947a38d3f6.png)
is divided by 
Solve for 
Therefore, 
![\[R(0) = 100(0^{100}) + 99(0^{99}) + 98(0^{98}) + \cdots + 3(0^{3}) + 2(0^{2}) + 0\]](http://latex.artofproblemsolving.com/0/0/4/0044c0eabc8eb4ba3d6b430ab908b3e92ba242f1.png)
![\[a(0)^2 + b(0) + c = 0\]](http://latex.artofproblemsolving.com/1/2/4/124caf8e3542d5c7e7d4b13b8324ad59758e5b8a.png)
![\[c=0\]](http://latex.artofproblemsolving.com/3/e/8/3e87824860754458838fc46dd729d700589ccbdb.png)
Substitute 
![\[R(1) = 100(1^{100}) + 99(1^{99}) + 98(1^{98}) + \cdots + 3(1^{3}) + 2(1^{2}) + 1\]](http://latex.artofproblemsolving.com/9/a/1/9a1757b781ed32cb6679971426b2ab28ecade5c7.png)
![\[a+b+c = \frac{(100)(101)}{2}\]](http://latex.artofproblemsolving.com/0/6/d/06d75968cb318133118fac2eac69df3e0c956e93.png)
![\[\boxed{a+b = 5050}\]](http://latex.artofproblemsolving.com/8/7/7/877ee0cf9be48f8dcee17815c0bd35deb3d3edb5.png)
Substitute 
![\[R(-1) = 100(-1^{100}) + 99(-1^{99}) + 98(-1^{98}) + \cdots + 3(-1^{3}) + 2(-1^{2}) - 1\]](http://latex.artofproblemsolving.com/0/7/5/075d546e826059f361650e03e721c5c148f65839.png)
![\[a - b + c = 100-99+98 \cdots -3 + 2 - 1\]](http://latex.artofproblemsolving.com/c/c/a/cca420756d22a06d25afe74d37d55465258c4888.png)
![\[\boxed{a-b = 50}\]](http://latex.artofproblemsolving.com/b/2/9/b292e0f9e4c3f42e823db2d4365bb991f5d172db.png)
Hence, 
and 
so 
Prove that
,
be a point on arc 
of circle 
.
and 
.
and 
be the reflection of 
and 
,
intersects 
,
intersects 
.
lies on 
Prove that
Let 
Prove that![$$ (a+b)(a+1)(b+1) \leq \frac{3}{2}+\sqrt[3]{6}+\sqrt[3]{36}$$](http://latex.artofproblemsolving.com/f/5/e/f5e68204507aa9c3e71c28844bbee3a804546b94.png)
Prove that
Let 
Prove that
Let 
Prove that![$$ a+b^2+c\geq \frac{135 \sqrt[3]{90}-2}{2}$$](http://latex.artofproblemsolving.com/8/6/6/866623c65f54d6147a43b1b241c2cba96863b624.png)
such that 
for all 
such that 
.
be a positive integer such that the sum of its positive divisors is at least 
.
distinct prime factors.
![\[{\ \{\frac{n}{10^k}} \} > \frac{n}{10^{10}} \]](http://latex.artofproblemsolving.com/0/c/a/0ca11fa76313b65fd76a3cceee90955960e44ef6.png)
.
is given. Find all functions 
such that for all 
we have
be an acute triangle, 
its orthocenter and 
to 
and 
to 
and 
and 
are the reflections of 
are the intersections of 
and 
respectively with the circumcircle of 
and 
lies on 
.
.
are positioned around the sides of 
Y by Eugenis, boompenguinz, hesa57, celestialphoenix3768, mathisawesome2169, DrMath, SenoritaJennifer, Ruby31, YouAreASnorlax, dlu200774, Halcyon09, CaptainFlint, Benq, coolprincess, rlybd5, michigo, AMC_Kid, TheMaskedMagician, Mudkipswims42, C-bass, srg12, mathguy623, IndianChina, 24iam24, Pythag314, TurtlePie, champion999, gamjawon, Adventure10, Mango247, Blue_banana4, Spiritpalm, PikaPika999, aidan0626
Hey guys, I made a Mock MATHCOUNTS test. Special thanks to hesa57 for proofreading, editing, and improving the test.
Submit your answers here, and shortly after, we will have your submissions graded. Submissions are due two weeks from now. That would be Friday, June 12th 11:59 P.M.
https://docs.google.com/forms/d/1rowZ_ky0P37fMzvRR6bIwE0y3rODtN1laRid2Caygak/viewform
Best of luck! If you see any issues, please PM hesa57 or me immediately. Thanks.
EDIT: The updated PDF is the last post I made on this thread.
Also if you do not know how to do some of these problems, after the test, you can ask on the forum.
Submit your answers here, and shortly after, we will have your submissions graded. Submissions are due two weeks from now. That would be Friday, June 12th 11:59 P.M.
https://docs.google.com/forms/d/1rowZ_ky0P37fMzvRR6bIwE0y3rODtN1laRid2Caygak/viewform
Best of luck! If you see any issues, please PM hesa57 or me immediately. Thanks.
EDIT: The updated PDF is the last post I made on this thread.
Also if you do not know how to do some of these problems, after the test, you can ask on the forum.
This post has been edited 10 times. Last edited by RadioActive, Jun 2, 2015, 8:13 PM
instead of finding the median of (1,2,3)....
or 
