MATHCOUNTS probability geometry trigonometry calculus LaTeX algebra

In the hard problem marathon i will post only one problem a week. It will normally require a lot of thinking, problem solving or its just simply too tedious or confusing. If you want to suggest any problems just pm me with Hard Problem in the subject line. I might be gone for a week in August so whoever sends the most Hard Problems that are posted on the forum can then send problems for the week(i will tell you guys when i am leaving) So whoever wants to join, just post here. By the way none of the problems will require anything except subjects you use in mathcounts (basic algebra, basic number theory, basic counting and probability, basic geometry) no trig calculus or stuff like that (because i am not very good at calculus  :D )

I'm joining..what's the deadline for the solutions to be PMed to you?

about a day to three days depending on how many solutions i get...

This looks interesting, i tentatively join

So am I. Make sure to keep the problems hard! :D

[quote="isabella2296"]I'm in, though I'll probably fail.  :roll:[/quote]

If you failed, I would fail..

I will join.
Question: Do we have to give a solution or answer?

i join, even though i'll probly be gone for a while.
probly solutions, shentang.

Anyone want to suggest any problems?  :D

here is the first one (this one is relatively very easy) :

Jimmy rolls 3 fair, six sided, dies. What is the probability that the sum of the number on the dies is divisble by 3 but not by 6?

Show solution not only answer

[quote="math154"]
Yongyi's solution was correct. However, being beast like he is, he got 10+25+1=45.  :wink:  :P[/quote]

 :blush:  :blush:

Too bad I can't think of an excuse to cover that up off the top of my head...

Does anyone want to post a new problem?  :lol:

no one will occupy anything!!  :rotfl:  :rotfl:

why won't anyone reply?  :cursing:  :furious:

This is not a tourney, you don't have to registrate.

[quote="herefishyfishy1"][quote="RisingFlame"]I'll post a problem.

$ 100$ chairs are arranged in $ 10$ rows and $ 10$ columns in a classroom. A student occupies a seat if and only if all the other chairs in the seat's row [b]or[/b] column are occupied. In how many ways can the chairs be occupied? (The order in which people sit does not matter, and some chairs can be unoccupied.)
Source: modified from Interm. C&P

EDIT: After you do the problem, generalize the answer for a classroom with $ m\times n$ chairs.[/quote]

Would it be $ 2^{10} \cdot 2^{10}$, or $ 1048576$? For $ m \times n$, it would just be $ 2^{m \plus{} n}$... wouldn't it? Because each row or column can either be full or empty...[/quote]

Nobody will occupy anything. First student wont occupie becouse there aren't occupied chairs, another students wont occupy for the same reason...

@herefishyfishy1: That over counts the arrangements where all of the desks are empty. I think it should be $ (2^{10} \minus{} 1)\cdot(2^{10} \minus{} 1) \plus{} 1$

[quote="math154"]Meh I just feel like posting a problem:
Factor $ a^4 \minus{} b^4 \plus{} c^4 \minus{} d^4 \minus{} 2(a^2c^2 \minus{} b^2d^2) \plus{} 4ac(b^2 \plus{} d^2) \minus{} 4bd(a^2 \plus{} c^2)$.[/quote]

I got this far: $ (a\plus{}c)^2((a\minus{}c)^2\minus{}4bd)\minus{}(b\plus{}d)^2((b\minus{}d)^2\minus{}4ac)$

New problem: There are two people playing a game on a nxn board. All the unit squares on the board are originally white, and the players take turns making unit square either red or blue. (Players do not have a set color: Player 1 could turn a square red and then turn a square blue the next turn.) After a squares color is changed, it can no longer be changed. The game ends when all the squares are colored. Then, player 1 wins if there is a 2x2 square where all the unit squares in it are the same color. Otherwise, player 2 wins. For which n will player 1 win?

(Problem inspired by the rated tourney.)


In a plane a set of $ n$ points ($ n\geq 3$) is given. Each pair of points is connected
by a segment. Let d be the length of the longest of these segments. We define
a diameter of the set to be any connecting segment of length $ d$: Prove that
the number of diameters of the given set is at most $ n$:

http://www.mathlinks.ro/viewtopic.php?p=347432#347432

In fact in the resources page there is a section for all IMO problems
http://www.mathlinks.ro/resources.php?c=1&cid=16


geometry
3D geometry
inequalities unsolved
inequalities

Let $x, y$ be positive real numbers such that $x^3 + y^3 = x - y$. Prove that $x^2 + 4y^2 < 1$.


calculus integration Euler derivative geometry logarithms symmetry

evaluate the integral of ln(sinx) from 0 to pi/2 using contour analysis/residue method.Help?

Consider $ \int_{C}\ln(1\minus{}z)\,\frac{dz}{z}$, where $ C$ is the unit circle. How is this related?

perhaps I should not have posted this here, because I am relatively new to complex analysis. I am looking for a solution. I think to answer your question, you used eulers formula, and then took the derivative to obtain the 1/z

No, that's not the process. I used my knowledge of geometry and the structure of the complex logarithm to get there.
Here's what you should do:
(1) Evaluate my integral $ I$ using complex analysis tools. There is a logarithmic singularity on the boundary, which doesn't cause problems, and a pole in the interior, which gives you the value.
(2) Parametrize $ I$, and write its real and imaginary parts as real integrals. One is zero by symmetry, and we don't care bout it.
(3) Manipulate the remaining integral until you get something involving $ \int_0^{\pi}\ln(\sin t)\,dt$. This may require some geometry.
(4) Solve for your answer.

I suppose I want to know how to do those steps. I am not sure of how to do them. also, I would like to know where your integral came from. I basically only know some tricks of complex analysis, and alot of notation.