Show that lim(x,y) -> (0,0) of $ \frac {2x^2y}{x^4 \plus{} y^2}$ does not exist but that along every line through the origin, the limit does exist.

through y = mx and the x and y axes I get 0 for the limit, but then how do I show that the limit itself doesn't exist?

Try looking along the curve $ x\equal{}t,y\equal{}t^2.$


geometry
quadratics
conics
parabola
probability
group theory
abstract algebra

I must share with you all this most magnificent exposition on the state of mathematics education.

[url=http://www.maa.org/devlin/devlin_03_08.html]Lockhart's Lament[/url]

I'm sure that most of the responses will just be praising it, but I'm more interested in hearing what people think can and should be done to remedy the problem, or if this actually is the problem.

My first comment as an ex-music major is that the first analogy is so ridiculously dead-on that I can't believe I never thought of it myself.

I hope you enjoy!!

I don't really recognize the illustrated picture, I have to say.

Swedish school system is pretty bad, and we don't learn very much. They've removed euclidian geometry completely because it is too hard, or something. Stupid socialist everyone-is-equal reforms. We don't have focused "courses" pertaining to a particular area like you do either.

Still on nationwide tests in grades 7-9 we had "open" questions. I remember two such test questions in particular: One was about how the dimensions of papers change in the A4-standard. The other was about how many solutions $ ax^2 \plus{} bx \plus{} c$ had. No hints for the questions, but a lot of time. The last question was unfair though, because quadratic equations is not something we learn how to do in those grades, but some teachers knew it was coming to the test so they taught their students about it beforehand. Anyway, I remember writing something about how it depends on the vertex of the parabola and then I tried to calculate the vertex but couldn't do it in the general case, and I still got decentish points on the question.

In high school I did the IB program, which is similar to american system, right? Well, I took Higher Level and it wasn't quite stimulating enough, but I still don't remember it being as boring and without place for creativity as the paper says.

What shocked me was how applicable that was to my experience of the British system!

I thought the comment about the quadratic formula hit home especially well, considering when a supply maths teacher tried to prove it, he failed and had to resort to me talking him through it!

I especially like the link between indoctrination and religion in the "syllabus" section.

[quote="Kalle"]... but I still don't remember it being as boring and without place for creativity as the paper says.[/quote]

I think a main thing to keep in mind when reading this is that it is half hyberbole and satire. The point is not that it is extremely boring, but that it would be so much more interesting if done properly.

I just have to say that you are very lucky if you could do a problem creatively outside of how you learned it and get credit. In America, almost every high school teacher I know can't reason whether or not a solution outside "standard taught technique" is correct. Thus, they mark something completely correct as completely incorrect. Hence why I identify with the very first analogy so well.

[quote="SimonM"]
I thought the comment about the quadratic formula hit home especially well, considering when a supply maths teacher tried to prove it, he failed and had to resort to me talking him through it!
[/quote]
I have a worse story. When we did group theory in high school my teacher did not know what "if and only if" meant. On (the centralized)  exams there were often questions where you had to prove "this if and only if that", but she couldn't tell me what it meant. Luckily my dad is a doctor in applied probability, so he could help me. Also, my teacher skipped many proofs which she thought were hard, even though we were required to know them. These were simple things such as a subgroup of a cyclic group is cyclic and Lagrange's theorem. If there were past exam problems she couldn't do she simply skipped them, so we were armed very poorly for the exams. Basically she only liked to write operation tables for finite groups and how to show isomorphisms between finite groups, leaving most of the topics for me to read on my own (and for the others not to read, and hence fail).

I agree with this pretty much wholeheartedly. From grade 0 to grade 10 I liked math because I was faster and more accurate at computations than everyone else, and it was enjoyable to be the best*. (Also, my dad is a maths teacher, and although he still teaches the curriculum, he has figured out some ways to make that stuff fun, so I actually liked doing computations.) Then in 10th grade, I read a book by Nahin about imaginary numbers, and it was [i]interesting[/i]. There were cool problems. He proved things. (Of course, I had no concept of mathematical proof at this time; to me, he was just explaining why.) If this hadn't happened, I'd probably have become a physicist or chemist (applying my skill at computations to something that was actually interesting -- discovering stuff about the real world). But since I discovered that maths could be interesting too and gave it a chance when I got to university, now that is my field of study!

My children are not going to learn math through our public school system. I don't care the details of how it happens. But they are going to learn to be artists, not calculators.

*(Actually, in grades 9 and 10 I wasn't the best anymore. But I was still very good.)

I couldn't have put this in a better way.

I remember being told to write proofs using the "Statement/Reason" format.  Generally things were needlessly stretched out, while it just seemed intuitive.  For example:

[asy]pair A,B,C,D;
A=(0,0); B=(70,0); C=(50,0); D=(120,0);
dot(A); dot(B); dot(C); dot(D);
draw(A--D);
label("$A$",A,S);
label("$B$",B,S);
label("$C$",C,S);
label("$D$",D,S);[/asy]
Given:
$ \overline{ABCD}$
$ \overline{AB}\cong\overline{CD}$
Prove:
$ \overline{AC}\cong\overline{BD}$

This is what it had to look like.  You'd lose points for not saying it exactly (or almost exactly) like this:

\begin{tabular}{l|l}Statement & Reason \\
\hline 1. \math{\overline{ABCD}}\text{, }\math{\overline{AB}\cong\overline{CD}} & 1. Given \\
2.\math{\overline{BC}\cong\overline{BC}} & 2. Reflexive Postulate \\
3. \math{\overline{AB} + \overline{BC}\cong\overline{CD} + \overline{BC}}\text{ or }\math{\overline{AC}\cong\overline{BD}} & 3. Addition Postulate \end{tabular}

I hated geometry proofs mainly because you had to write every single thing down.  Something as trivial and intuitive as this needs to be proven rigorously.

Yes, we were able to take some shortcuts.  Once we proved a "big" theorem (such as a right angle being inscribed in a semicircle, the relationship between the angles formed when two parallel lines are cut by a third one, etc.), we could just cite that.  It was still tedious, because something that seemed intuitive required to be proven "over-rigorously."

My current teacher is better.  When discussing a theorem, she actually writes out the full proof (except for the formula of a hyperbola due to time constraints).  She proved the formulas for the conics (at least just for those where $ b = 0$ when put in $ ax^2 + bxy + cy^2 + dx + ey + f = 0$ form).  Although I don't recall her actually showing that the conics are formed by cutting two cones with a plane (and maybe she did -- I forgot), it's a start.  She says that she is showing the theory behind the proofs, and not just showing which buttons to push on the calculator.

If I were ever to become a math teacher, I would teach in the style that Paul Lockhart sees as best.  In fact, that's how I believe mathematics should be.  It may be harder to understand at first, but once you catch on, you will soar.

When my sister was in 5th grade, she had to do these kinds of problems:
$ \Box\Box\Box\times\Box\Box= 15974$
Fill the numbers 2, 3, 4, 6, and 9 into the squares using each number once so that this equation is true.
The class was expected to do this in a guess-and-check manner.  Yeah, the chances of guessing it right on the first try are $ \frac {1}{120}$.  My mom taught my sister to approach this logically, with methods such as looking at the units digits, i.e., working in mod 10.  Everyone was blown away by her explanations.

And you're going to love this one:

When I was in 5th grade, we were taught "Russian Peasant Multiplication."  Here's an example:

$ 83\times 24$
\begin{tabular}{c|c}83 & \color{red}24\color{black} \\
41 & \color{red}48\color{black} \\
20 & 96 \\
10 & 192 \\
5 & \color{red}384\color{black} \\
2 & 768 \\
1 & \color{red}1536\color{black}\end{tabular}
Where the row that comes after $ a|b$ is $ \lfloor a/2\rfloor | 2b$.
Finally, you add the numbers in the right column where the number directly to its left is odd (i.e., the red numbers) to get the desired product, which is $ \boxed{1992}$.

This would've been a cool thing to just mention, but we were actually TESTED on this!  We weren't even told why this works!  I understand why it works now, but still, teaching this was pointless and stupid!  What exactly is wrong with the classic method that we all know and love?  Much less time consuming, and it takes up a fraction of the space on a piece of paper.

This is not the kind of material that prepares you for MATHCOUNTS (hey, it's never too early).  I guess I can see why this was taught, because I don't think my school has even heard of MATHCOUNTS.  I sure wish I knew about it back then.  I would've pestered the middle school math department to get involved in that.

The only thing I don't like is the emphasis on how math is useless. Math can be enjoyed while being used. (In fact, one could argue that math is superior to other art forms for the reasons of its usefulness.) Sure, some concepts are too abstract to be used, but then again, some aren't. He argues that all useful math can be done with a calculator or computer program... but there is some applied math that is very nontrivial. Also, who makes the calculators and computer programs?

Everything else I pretty much agree with. I especially like the analogy in the beginning.

I also like the part where he says that people who think they are good at math find out later that they are actually just good at following directions. Very true.

(Note: I didn't read the whole thing.)

[quote="tjhance"]The only thing I don't like is the emphasis on how math is useless. [/quote]
I don't think he means that at all. I think he means that abstract math is to real-world math as Michaelangelo is to painting your ceiling white with a paint roller. You're doing the same thing, in some very general sense, but you don't need to take a class to figure out how to paint your ceiling, whereas you do need to learn about art in order to really appreciate what Michaelangelo did. However, people tend not to think of math in this way in our society; they think of math as just the thing that you do in the real world, like balancing your checkbook, and they try to teach math that is specifically aimed at the real world. This is why, in high school classes, you end up doing these absurd problems like "A guy is walking away from a 10m tall lamp. He's 2m tall and moving at 1m/s. How fast is the head of his shadow moving when he is 20m from the lamp?" There's actually no reason to do this in real life! But for some reason people think they they need to teach everyone to be able to solve problems like this! It's ridiculous.

Before I read this, I could never figure out why math could be so fun while writing a page of 2-column proofs was torture. Now I see why the proofs taught in geometry are so hated. After students learn to do a lot of hard problems themselves, following the curriculum, no matter how advanced, simply seems [i]wrong[/i]. In fact, it horrifies me that Lockhart described my geometry class so accurately, including the details about the textbooks. The "real life situations", which the books call "problem solving", is even worse. How much more interesting is it to prove a parallelogram on a folded music stand?
     I agree that mathematics is an art. I felt that way ever since I was introduced to my own chances at problem solving. Although I do not know an effective way to remedy this epidemic problem, I believe that there are some teachers who DO want to share true mathematics with their students. Perhaps we could loosen the curriculum by emphasizing more self-exploration of a topic before it is formally introduced, and adhering less to the textbook's repetitive questions. I enjoy reading AoPS 2 much more than school material even though I don't understand all of it because I meet interesting approaches to solving harder, more abstract problems from the former. Reading this made me HATE how "those who catch on to patterns faster are considered good students". It's a shameful reward for being docile to the "curriculum".[/i]

[quote="jeff23"]I don't think that's a worse story. If you did abstract algebra in high school, then it is completely understandable if the teacher doesn't understand it. I would say most universities (in America) wouldn't expect their students to know how to prove those things on a test -- forget about high school! I'm almost tempted to get my kids educated in Sweden if I ever have any.[/quote]
As I said I did "International Baccalaureate" in high school, so you can enroll your kids to high school where you do abstract algebra (not all schools offering IB offer math higher level and math further studies, though).

About this "self-exploration", this isn't really done at university either until you start writing papers and similar things. Should the way mathematics is taught in university also be changed? I have a hard time seeing how you will get to all the essential theorems if you spend time self-exploring explored things ...

[quote="zephyredx"]The "real life situations", which the books call "problem solving", is even worse. How much more interesting is it to prove a parallelogram on a folded music stand?[/quote]
One of my former math teachers said a very similar thing.  He said that the state tests use questions that sound like poor attempts to create real-world applications.  It's rather annoying, in my opinion.  Not only are there much better ways to make real-world applications, but mathematics, as said before, is an art, and should be appreciated and taught as such.

[quote="Kalle"]
About this "self-exploration", this isn't really done at university either until you start writing papers and similar things. Should the way mathematics is taught in university also be changed? I have a hard time seeing how you will get to all the essential theorems if you spend time self-exploring explored things ...[/quote]

Usually universities call this the Moore Method, and it is pretty common in analysis and topology intro classes. That is one of the trade-offs that you have to weigh in deciding to teach a class in this fashion. Most professors I think rationalize it by saying that by learning using the Moore method, you will be able to learn all the major things you missed and the details in a matter of probably a week after finishing the course due to the much higher level of understanding.

All of a sudden I feel incredibly inspired to become an elementary school math teacher.

[hide]http://www.mathlinks.ro/viewtopic.php?t=193507[/hide]

This brings everything back into perspective. It is all well and good to want teachers who understand that maths is more than plain computation and formulae, but seriously can that really be a teacher?

One problem is that a school may not have the access or the money to hire so many high-quality math teachers.  Such teachers may only be used to teach courses such as pre-calculus or calculus, and are not even seen by students who just don't care about mathematics.

Generally, the BEST (not on a classroom level, but on a statewide, national, or possibly international level) math students don't usually become math teachers.  They usually go on to higher things.

I would like to teach mathematics someday (possibly in my later years), and I would want to teach it as an art form.  The only problem is that this may be too radical, and it could cost me a job.  But hey, maybe the math curriculum will change in the next 20-30 years.  And that's a pretty big maybe.

[quote="lingomaniac88"]Generally, the BEST math students don't usually become math teachers.  They usually go on to higher things.[/quote]
[i]This[/i] is the real problem. Our culture views teaching as pretty "low". Most people consider high school teachers to be the people who couldn't cut it in industry or academia, plus a few weirdos who actually are passionate about teaching and believe that it's the right thing to do with their skills (and hopefully your kids will be taught by those weirdos).

Before we can have really good grade school education, being a grade school teacher needs to be an illustrious profession. But that would mean teaching salaries would have to be competitive with professional salaries in the same field, and this is well beyond the reach of our current social and governmental system. (We'd either need a LOT more tax money to go to schools, or schooling would have to be much more privatized and competitive.)

Is anyone here from East or Southeast Asia? What is it like there? I know that education is considered a much higher priority in the culture there, and it's extremely competitive to get into "good" schools, even middle schools. Is teaching at a good school a prestigious job? Are the teachers at the good schools actually much better, or is the quality of the school just a reputation, empowered by government mandate, tradition, and selection of students rather than quality of education?

[quote="Xevarion"]
Before we can have really good grade school education, being a grade school teacher needs to be an illustrious profession. But that would mean teaching salaries would have to be competitive with professional salaries in the same field, and this is well beyond the reach of our current social and governmental system. [/quote]

I can't speak for others, but I get a general impression (and I'm in this school of thought) that people who learn higher mathematics are not shunning teaching due to money. I could probably teach how to solve $ x\plus{}5\equal{}7$ at $ \$100,000$ a year for about five years before the money wouldn't be worth it. Now if I were teaching "real" math, then the current salaries wouldn't be a problem. It truly is content for me.

I think it's naive to believe that people make their career decisions about what they actually want to do. Most people don't actually know what they want to do, and many that do still make decisions based on money anyway (even if they don't do it for themselves, maybe they do it because their families need it, or their friends or family pressure them into it).

I think at Princeton, about half of the math majors end up working for finance companies, and all three of the aforementioned reasons are well-represented.

Consider the following: I know that even if I choose to work for a hedge fund, nothing will be able to stop me from thinking about math and keeping up with current research in math. When I'm retired at 35, I could be an "amateur mathematician", or try to go back to graduate school. If I have children (which I hope to), I'll certainly be able to teach some kids math, even if I am not a grade school teacher. (I could also be a part-time tutor.) So I really don't have to pursue a career in mathematics in order to do these mathematically fulfilling things, like thinking about math a lot or teaching people math (at any level). If I also like working for a hedge fund, what's to motivate me to pursue math anyway? Nothing....

This lament is basically everything I've been thinking since pre-school. Except for the slightly above vocabulary and metaphors. Everything else was pretty much the same, you know?
Oh--and I was one of those students who were good "instruction followers"  :lol:

[quote="Xevarion"]
Is anyone here from East or Southeast Asia? What is it like there? I know that education is considered a much higher priority in the culture there, and it's extremely competitive to get into "good" schools, even middle schools. Is teaching at a good school a prestigious job? Are the teachers at the good schools actually much better, or is the quality of the school just a reputation, empowered by government mandate, tradition, and selection of students rather than quality of education?[/quote]

Well, I can tell you that education is a higher priority there. Let's just say that more American students lose sleep over school than, say, Chinese students do. They take exams for the middle schools, and if you score at a certain cutoff, you get in for a normal amount or for free--I'm not sure which, then if you score below that cutoff but above another one (generally 8 or so points below), you pay a certain amount for every point to raise that score to the first cutoff. The fee can be extravagant, tens of thousands of [insert currency here] for one point. That's pretty bad. Then, below the second cutoff, no matter how money you have, you can't get it. Period. It used to be much worse--and current times have also remet with the college entrance exams. Now [i]those[/i] are intense. With really hard, like IMO, give or take a few degrees of difficultly like problems. As well as other types. :P Teachers used to also be much more connected with students, really one-on-oneing and posing challenging problems. It's gotten worse, as well. Nowhere near other places, but slightly worse, yes. Oh--and this stuff all pertains to China. Should've meantioned that earlier--but nah! I'll make you read the whole thing. :P

I guess I can speak a little bit about high schools in Hong Kong and China.

In Hong Kong, there are two kinds of schools with more fame. One is traditional prestigious school, whose fame alone is able to attract many good students graduating from primary school each year. Teachers there in general are not quite able, in the sense that they probably know their stuff, but their teaching aren't really good/motivating that makes the students feel that the subject is interesting. However, due to the quality of students and the culture of tuition schools, these schools are able to produce students with excellent academic results in public exams on yearly basis.

Another kind of schools are "locally famous" schools. They may be famous in a district, but the fame cannot be compared to the fore mentioned type of school. This type of schools usually have better teachers, whose teaching enables students to do well in public exams, which in turn builds up its fame.

In general, the whole atmosphere in Hong Kong is about money making. Teaching is a job with high workload/pressure, while they don't really make much money. I don't think it is a prestigious job here, except for those who teach in tuition schools. Tuition schools prepare students to tackle exams, not learning. They write up summaries of a subject, they give us the past papers/mock papers of public exams and so forth. Attending tuition schools is a very dominating culture in Hong Kong, and the teachers in tuition schools are shaped like superstars. They are making much money.

----
I don't really know much about education system in China since it is very different from that in Hong Kong. I believe that they have two public exams in high schools, one (zhongkao) after grade 9 for admission to high schools, another (gaokao, something like A-level) after grade 12 for admission to universities. Chinese students in general spent much much time on the A-levels. I heard that in grade 12 actually there's no teaching, the students actually keep on reviewing the materials, doing past papers and practice tests and so forth. It seems that teachers in high school play a much more influential role (comparing to Hong Kong) in the preparation of A-levels of students. Also, teachers with good teaching are rewarded. The better ones are called "star teachers" (teji jiaoshi). They also play an important role in students' preparation to various type of competitions, like olympiads in math, informatics, physics and so forth. So I suppose teachers at good schools are really much better.

At my school (a private school), in eleventh grade (well for most kids...), we have Precalculus (one "regular class", and one "honors" class").  The interesting thing is, PCBCH (the name for the honors class) is not a class just designed to teach you more concepts, but rather one designed to open up the vastness of mathematics to you, and show you how to think creatively about it.  Then after that, there's an optional class (often taken with calc or after calc, usually has about 10 kids per year) called Math Seminar, which is all about exploration (for example, the topics vary from year to year, usually including inversive geometry and number theory).  Pretty much every student I know who's taken one of those has suddenly changed to love mathematics, and has told me those were the best math classes they'd ever taken.  Perhaps it has to do with the fact that we have a couple of teachers teaching those classes (particularly the head of the department, who is like my mentor) who are really passionate about math and love to do it in their spare time...

Anyways, it helps to have good, passionate teachers who are given the freedom to teach what they want...obviously this might not work so well in a public school...

Lockhart's lament is pretty much dead on from my experience. I don't know how feasible it is to cover all the important basic theorems using the math-like-art approach, but considering how much time we waste in elementary and middle school as it is, I would like to see that approach tried anyway. (It would seem that it has worked for Lockhart at least.)

In Brazil, we don't have proofs on geometry. It seems we don't have geometry at all. Only the basic definitions and the so hated formulae on basic school. In high school we don't have geometry as a specific subject. It appears only in the beginning of trigonometry, and resumes to calculate a side of a right triangle given the sin of an angle and another side. I was a good monkey, and always obtained good results. I knew true mathematics when I started participating in the Brazilian Math Olympiad. Now I struggle to solve those beautiful problems. It's so good to get frustrated :P .

I think Lockhart is somewhat radical. The basic arithmetic and basic (I say BASIC) notions of algebra ($ x\plus{}5\equal{}9$ find $ x$) are fundamental to life and have to be taught. I agree when he says that we need to understand what mathematics really is, and to present it to our children as an art an as something that could be fun.

I absolutely agree with what Lockhart said. However, I think he comes across as somewhat radical, and I am somewhat biased, since I am the only one really into math in my school. But I think the lack of creativity in mathematics education is because if math was taught that way, good grades can't be easily forced by a hard-working student if they are not naturally good at it. So there are two sides to the argument. For example, the kid who is top in my class and always got As in math could only solve 4 AMC10 problems. When I showed the kids in my class how to do some of the problems, some of them called the test "stupid". But then you have to look at it from the other side too. If a kid works really hard, I don't think it's right to disappoint them with a B, let's say. So in the end, I guess it's just best to keep the education system the way it is and leave competitions for the other kids.

By the way, I was cracking up when he wrote: The area of a triangle is equal to one-half its base times its height. The question has been asked and answered at the same time— there is nothing left for the student to do.
Also when he wrote about the vertical angles proof: Instead of a witty and enjoyable argument written by an actual human being, and conducted
in one of the world’s many natural languages, we get this sullen, soulless, bureaucratic form-letter
of a proof. And what a mountain being made of a molehill! Do we really want to suggest
that a straightforward observation like this requires such an extensive preamble? Be honest: did
you actually even read it? Of course not. Who would want to?
The effect of such a production being made over something so simple is to make people
doubt their own intuition.

This reminded me of my own experiences as a kid, following the standard math curriculum.

In third grade, I remember how I was proud of myself because I knew the multiplication table up to the 12's while everyone else was having trouble getting to the 9's. Also how I knew division and fractions while my classmates were working on single-digit addition/subtraction.

In fifth grade, taking pre-algebra, I remember learning a formula and asking my teacher why it "worked" (at the time, I didn't know it was something called "intuition," my preferred way of learning). He wasn't able to do anything more than derive it algebraically.

In sixth/seventh grade, I began to realize how contest math was much more fun than school math, especially the geometry class I was taking. I tried hard to test out of the class (successfully) because the two-column proofs were boring me out of my mind. Seriously, you don't need to "learn" CPCTC to know it's true..

I cruised through algebra 2, but by eighth grade, I began to lose sight of real math again. Now, having finished calculus BC, I feel that I've skimmed over the material way too fast. Even though I have a much more intuitive grasp of the material than do most of my school peers (eh am I being too presumptuous?), I'm not much better off.

I wish Lockhart was my teacher. :P

The dead-on-ness of the article was so shocking and awakening... finally something that is actually true! It kind of made me jumpy and excited (in a way that is rather embarrassing) for the first 5 minutes after I read it.

And then after that you feel incredibly depressed (at least I did). It seems to me these posts were made within the first 5 minutes after reading it.

And to think that the whole world has the same attitude (I'm thoroughly convinced that there is no magical land of Romania or Bulgaria or China or Korea or Math-land where the majority of students view math in the same way as Lockhart does). It would take an unimaginably great amount of resources and whatnot to realize these - and I am sad to call them this, but they are - dreams.

That article was the best I have ever read. It's everything I wanted to say put into words.

I've always sensed something wrong with my math classes in general. I had a great tutor in fifth grade that taught me quadratic formula, factoring, and even went a bit into calculus o.0. Of course, at that time I had no idea what math was even about. For years, to me, it has just been another "subject" to be "studied". However, the practical applications of English and Social Studies are pretty much obvious, and even though I was <relatively> "good" at math, I had no idea what it even was about. It just felt meaningless, to learn something and then not use it. The "usability" of math idea was with me until a little later, until I decided to just forget it and, as Lockhart says "have fun".

I've always liked jiggling with variables, especially in geometry. Variables and Pythagorean Theorem can do wonders. In 6th grade, during my very first AMC 10, I forgot the formula for the area of an equilateral triangle. I derived the formula during the test.

In 7th grade, my best friends and I all made the math team, and we were given a "packet" of formulas to remember. This was when I began to see mathematics in a new perspective; It is fluid, logical, and awesome (I have no words to replace awesome with.)

Still, remembering formulas is tough, deriving them is so much easier.

Finally, this summer, after some 7 years of education, I finally understand what math is and what it can do, and I agree very little of it is actually useful. Geometry, Trig... We have computers and protractors. We can measure with a tape measure instead of "using law of cosines over AB". But, honestly, I think it's more interesting to take the law of cosines over AB.

Agreeing with Lockhart that mathematics is an art and should be taught as so. I am haunted by thoughts of doing ALG I in an enforced way (Write the equation, skip a line, subtract 2x from both sides, skip a line, divide by the coefficient 3, skip a line, find the value of x, skip a line, circle it, move down, check your answers by substitution...) basically, I end up covering the sheet of paper with "various solved values for x".

I've never thought that you could create a whole new arithmetic by taking remainders... Or that the sides of a triangle are in some way proportional to the triangle's angle measures... Or that exponents could be used to calculate factors...

What Lockhart said is true. (I feel so sad now). Many people I know would be really interested in "mathematics" if they knew what it was. And education doesn't give them a chance, society keeps them from changing that. I actually believe that the concepts would be learned faster if the students were left on their own, with a few, maybe many, prods (but no lectures). I've wondered what use is vocabulary anyways.

Not to say math is totally useless in today's world, but we leave that to the scientists  :D .

The following is pointless ranting/praise (of the article) to some degree, but this article was awesome.

My favorite sentence:

"Everyone knows that poetry and music are for pure
enjoyment and for uplifting and ennobling the human spirit (hence their virtual elimination from
the public school curriculum)"

I do math for that exact reason: pure enjoyment and for uplifting and ennobling the human spirit. I also wish that I could not take a math class for the rest of high school, since I am like, learning nothing in class.

The analogy is great too, especially the part about how Michelangelo had other things on his mind when making the painting for the chuch ceiling, and how Beethoven could make an ad jingle, but chose to do harder stuff (I only practice hard music too)
Also, the great analogy helped me to understand why math is the "music of reason". (I read the Music of Reason book by Theoni Pappus without understanding why math was the music of reason)

I also liked how he describes math as fantasy - I spend like all the time on the school bus fantasizing about random stuff, and now I finally realize that I fantasize because I need a break from the ridiculous schoolwork I have to do.  I guess I do math because I love to fantasize and explore (I never solve the problems I make and explore.....they are too hard, yet I explore nonetheless)

Also, this article points out that we should explore a question before we a given an answer for it. That's why the art of problem solving books are so great - they start off with questions.

Also, I like how high school geo proofs were called the devil's work: nothing could be truer.  I once thought that for olympiad problem, I would need to state the theorem that states that $ \angle A$ (which my geometry teacher said was the GEOMETRIC FIGURE OF AN ANGLE ITSELF)$ =m \angle A$ ( which my geo teacher said was the MEASUREMENT OF THE ANGLE). However, now I realize that if I asked a USAMO or USAMTS grader whether I would need to state this trivial fact (this is actually not trivial, its actually POINTLESS in the world of mathematics) in a proof, they would not even know what I am trying to say.
The other thing that was the same as this was the "theorem" stating how $ AB=AB$ (the second AB should have a line over it, I have no $ \text{\LaTeX}$ skills), where $ AB$ was apparently the LENGTH OF THE LINE, and $ AB$ (with a line over it) was apparently THE GEOMETRIC FIGURE ITSELF. Hm, I am glad to realize that I have almost completely forgotten this.

Once again, sorry for my ranting, but I totally agree with this article

Once, in 6th grade, my math teacher told me that a triangle with angles 30, 30, and 120 was not necessarily isosceles. I proved the fact to her. She said, "well, you must be wrong, since the answer book says it's an obtuse triangle.

I think Lockhart hit the nail on the head in his "true and honest math curriculum." Next year, I will be taking Geometry, and, having glanced at those disgusting mutilations of something so beautiful, am now feeling slightly claustrophobic. (I have already done vol. 1 and 1/2 of vol. 2, so I can already do all this in a much clearer and more intuitive manner.)

Once, I asked my teacher why the Pythagorean theorem works. I was treated to a lengthy discussion about how, in mathematics, it was important to follow directions and be meticulous. Whatever happened to creativity, to tinkering, to writing numbers out and seeing what happens, to intuition, to true problem solving, to freedom of thought, to unfettered discovery???

Being an AoPSer makes school math even more tortuous to me. I've been trained to sharpen my ideas with intuition and logic, not to dull them though bureaucratic formulas and postulates.

Once, a "good math student" walked up to me and announced "the square of the hypotenuse is equal to the sum of the squares of the legs." To this, I calmly replied, "prove it."

Sorry for reviving an old thread, but I'd just like to throw my ideas out there.  Personally, I think Lockhart's Lament is not only an amazing piece, but I would go so far as to say that every, [b]every[/b] math teacher should read it before they begin teaching, and it should definitely be mandatory for all math students to read sometime in their school career.  It effectively summarizes every ideal that the AoPS community stands for.  I know that school is "important," but I'm sure any AoPSer would agree with me that math class is for students whereas imagination is for mathematicians.  And I am willing to bet anyone on this planet that I have better grades than Einstein ever had and that I can draw music notes better than Mozart ever could.  Alas, such trivialities mean nothing in the realm of intellectuals.

I'm personally a casual gamer, and because of that I'm pretty lazy. So, when I went into Geometry class, (I'm in eighth grade) it was like good... nice... finally I won't fall asleep in class. I proceeded to do my homework, and I GOT MY CALCULATOR CONFISCATED just because there were tons of calculations, and my teacher insisted that those were legitimate math. (I wish that there were more trigonometry problems. Then I get away with using calculators.)  I was like WT-?!! and then I resorted to doing my homework at home so I could use a calculator. This was at the beginning of the year, and I still use calculators as a crutch, and I don't really care because there is no need to practice arithmetic when you are already at a very high level like Geometry or above. Then, it came to the proofs, and then it was absolutely boring. In fact, I fell asleep during many of the lectures (I am actually the smartest in the class.) because I don't want to do like 30 problems of arithmetic for homework. Reviews- expect me asleep or listening to my MP3 player secretly. Not exactly a role model for the average student, but I just do that. Granted, the teacher calls me "special", but that is not even the way that it is supposed to be. Rather, I think that it is THEM that are special. Heck, here is one of the problems from my geometry book-
---
You are planning a vacation to Montana. You want to visit the destinations shown in the map.

A> A brochure states that the distance between Granite Peak and Fort Peck Lake is 1080 KM. EXplain how you know that this distance is a misprint.

B> Could the distance from Granite Peak to Fort Peck Lake be 40 KM?
---

and then so forth. I was thinking about like the traveler's problem, but no... it was just about the same question... repeated multiple times.

I have a friend in the class who also hates classroom math but has ambitions elsewhere. So, he nearly always BSes through homework. (Naughty, but we're only here for the credit.) In fact, I always do BETTER on out of school tests than on actual math exams given at school because of the fact that it is a different type- thinking is involved.

The only word that I think should be used in mathematics is memorize or a similar derivative. Make sure that your teacher does not say "Memorize and then you will do well." Big whoop.


geometry
3D geometry
sphere
geometry unsolved

Three circles of radius $a$ are drawn on the surface of a sphere of radius $r$. Each pair of circles touches externally and the three circles all lie in one hemisphere. Find the radius of a circle on the surface of the sphere which touches all three circles.

consider the "poles" of each circle (which are points on the sphere) as $O_1,O_2,O_3$. Easily distance between each two of these points is $a$. Now consider the tetrahedra formed by these three points and the center of the sphere. $T$ the center of the face consisting those three points has distance $\dfrac{a\sqrt3}{3}$ from them. so $\angle{ O_1OT}=arcsin(\dfrac{a\sqrt3}{3r}$. If $K$ is the tangency point of $C(O_1)$ and the desired circle, $\angle{O_1OK}=arcsin(\dfrac{a}{r})$. if we subtract these values, we find $\angle{KOT}$ and then radius of the circle


Dear Members,
                      In 'Nash Equilibrium,"the game is started by two,
or more players in opposing sides.Each of them had their own
set of strategy that will fit to a particular given situation.Such
given strategy may,or may not be known to each of them.In his
famous booh"The Strategy Of Conflict,"2005 nobel prize winner in
economic "Thomas Schelling" had invented an idea known as"Focal Point."Both of these two game theorists worked in the
1950's think-tank "RAND."Both of them in all their works assumed
the following:
             1)The exixtence of player,or players engaged in a certain conflict against one another.
             2)The areas of conflict wherein the game is to be played.
             3)Plan of each other in which each player,or group
of player's strategy may,or maybe known,unknown to them.

                    Would it be possible to play a game in which  the
conflict at hand is usually at the hand of the oppossing side whose existence,or way of communication "continue to change
in suucessive scenarios but,still "UNKNOWN" to the other side.
(Unknown means:Not knowing player's way of diplomacy,communication,even coomunication.Onll way to know
their strategy is their pasts actions related to their collective,
or,single action. :(

                   I 'm talking in terms of using "mathematical abstractions"to deciphere the future,probable move,strategy
of the oppossing sides. :?: Such is a mathematical description
related to "Game Theory Forescasting." Can we not attached
a kind of "variable"whose value is connected to the past event
or action of both antagonist? : :|

                                                       Sincerly,
                                                     Mikhail S.

In order to generate an effective game-theoretic strategy, you absolutely should view your opponent's responses as a variable with stochastic inclinations, leaving open room for altering the parameters under which the variable is still stochastic.  For instance, if a player chooses "rock" 40% of the time while playing rock, scissors, paper, the variable should be adjusted.

Other adjustments can get very complicated.  Somewhere on the web there is a description of a rock, scissors, paper contest played by AI programs.  One of the recent winners of the competition was a program called "Iocane Power" (in reference to the movie [i]The Pricess Bride[/i]) due to its depth of variable adjustment in terms not only of the way it detected patterns in its opponents, but its suppositition that its opponent was doing the same thing, it's suppositition that its opponent would know that Iocane Powder knew that and would make Bayesian adjustments of its own, etc.

Suppose the "odds bet" for Boston to win the World Series is 3 to 1.  If you bet $x on Boston, then you pay$x if Boston doesn't win while you receive $3x if Boston does win. Matt is an expected - utility maximizer with utility function u(x) = lnx and initial wealth$60,000.  If Matt believes that the likelihood of Boston winning the World Series is 1/1, how much would Matt choose to bet on Boston?

Infinite money, if he "knew" that boston would win  :P


Not going to post the answer straight out, so MathCounts students will be able to answer this on their own.

[hide]$x^3-x^2-4x+4 = (x-2)(x+2)(x-1)$
$x=2, x=-2, x=1$[/hide]

[quote="chess64"][hide]\[x^3-x^2-4x+4=0\]
We can use guess-and-check synthetic division. Let's see if it's divisible by $x-1$.

1 |  1  -1  -4  4
	|__[u]1[/u]_[u]0[/u]_[u]-4[/u]___
      1    0  -4   0

Sorry, it's hard to do synthetic division on this.

We see $(x^2-4)(x-1)=x^3-x^2-4x+4 \Rightarrow (x+2)(x-2)(x-1)$, so $x = \{1, -2, 2 \}$
[/hide][/quote]

You don't have to use synthetic division.

I like using synthetic divison for quadratic functions of higher degree. It's much faster than factoring the function.

Cheating:  Use a graphing calculator and plug it in--that's what I did...I'll post some screenshots later.


if i have an expression

$ A\equal{} 5sin(377t\minus{}40^o)$how do u know whether to add $ \plus{}90^o$ or $ \minus{}90^o$ to change it to an expression in cos!

When in doubt, derive!

$ \sin(\theta\plus{}90)\equal{}\sin\theta\cos 90\plus{}\cos\theta\sin 90\equal{}\cos\theta$.
$ \sin(\theta\minus{}90)\equal{}\sin\theta\cos 90\minus{}\cos\theta\sin 90\equal{}\minus{}\cos\theta$.


Putnam
topology
vector
algorithm
quadratics
graph theory
number theory

Is it possible for one to be extremely good in math (i.e. does good on the putnam, good at proofs, topology, etc..) but awful in programming? In other words, would a PhD mathematician in pure math need no programming skills? Also can one be a very good programmer but a bad mathematician?

Thanks

A very good math student has good logic and he "sees things" (like in proofs).  In programming, you are faced with a problem and try to make a program to make it easier or get rid of the problem.   A math student would see how to come about doing this very easily.  Thats my opinion.

[quote="ragingbullrocky"][color=red]Is it possible for one to be extremely good in math (i.e. does good on the putnam, good at proofs, topology, etc..) but awful in programming? In other words, would a PhD mathematician in pure math need no programming skills?[/color] [color=green]Also can one be a very good programmer but a bad mathematician?[/color]

Thanks[/quote]

[color=red] This is very VERY unlikely. Math requires certains skills of logic and structural visualization(i have no idea what that is) that makes programming much easier. [/color]

[color=green]I know for a fact that this is possible. My AP computer science teacher was the IT manager at some relatively big tech company. He originally wanted to be an electrical engineer, but he decided to change his major because he coudn't pass calculus. He knows most programming elements closely related to math, but class isnt as fun :( , because, that kind of stuff is the most fun[/color]

[quote="ragingbullrocky"] In other words, would a PhD mathematician in pure math need no programming skills?
[/quote]
The answer to that is "yes, you can make a brilliant career in pure mathematics without having the slightest idea of how to program nowadays no less than 60 years ago". On the other hand, I do not think that any good mathematician would not be able to master basic programming skills if he really wants to learn them. Whether this statement is true the other way around is unclear for me: of course, there are many programmers who have difficulty with basic mathematics, but the best programmers I know are all mathematics or physics majors who just decided not to pursue a career in pure science but were among the top 10% as students.

Also, great mathematicians like Ptolemy and Pythagoras didn't know programing language,.  :D

yeah, but it seems incomplete for a mathematician to not know how to program.

In my opinion, any decent programmer needs to know a good amount of math, as it pops up everywhere. For displaying objects on a screen, you need to be very comfortable with vectors, for one thing.

For more advanced examples, you're going to need to be pretty familiar with lots of proof techniques if you are going to prove that a greedy algorithm (or algorithm in general, for that matter) works. It also helps you determine the runtime of your algorithm. It helps a lot with dynamic programming and recursion. Graph theory is common to both math and computer science. Number theory and functional analysis are also two fairly common computer science topics. Linear algebra is often employed, and can sometimes reduce a program by a factor of $\frac{\log(N)}{N}$ in runtime. Case analysis, though not strictly mathematical, is very handy. Calculating probabilistic bounds requires math. I don't even know that much computer science, so the list probably goes on and on.

I don't think it is as important for a mathematician to be good at computer science, but it would probably be fairly easy to pick up, and can be useful (to an extent) in finding patterns, though you often get more out of discovering the patterns by hand.

Well, considering the fact that u need to score atleast 80% in Math to be able to take up Computer Applications(After STD X here)...and that u need math in almost all the programs u do...I guess if u gave Pytahgoras and Ptolemy computers and taught them programming, they'd be good at it!! I certainly think that both are very closely related...

I graduated theoretical mathematics, but I had Programming (Pascal, Fortran) and Advanced programming (C++, Lisp, Prolog) as obliged exams. It was relatively simple to learn those exams, but I can't say that all mathemaicians are excellent programmers. It is obvious that mathematical thinking and analysing of any problem can make its solving easier, but I think that fair amount of talent for programming is needed if one wants to be good in programming. Many of my colleagues who graduated Programming & informatics don't dare to find a job in comanies where programmers are needed and they work in schools as teachers. Main reason for this is that they had too much math and not enough programming during their studies. The other problem is our terrible school system which is completely out of date and very far from real life requirements.

[quote="Nienna"]I graduated theoretical mathematics,... [/quote]

:) I graduated theoretical physics, so I'm not a mathematician but I hope I'll also have a vote. I had Fortran, C++ and Mathematica, and also think that it was relatively simple. Some of my partners who were among the best of the physics class however didn't think so. So I would say that there are good physicists who do not master computer science. I even have this friend who enjoys the most axiomatic approach to physics which is rather mathematics than physics and is truly bad at programming. He is also not interested. I think that if you put to it, you learn, and programming is, as physics or maths, about learning how to think. I myself can now program C++, C, bash, Fortran and I'm beginning to learn python. Once you understand one language is increasingly easy to learn others.

Ursula

Ursula, in my opinion it's more accurate to say, once you learn one paradigm of programming, it's easy to learn languages which allow you to program using that paradigm. Switching from C-type to Haskell-type programming for example, can be difficult.

I like to think of computer science as a subset of mathematics. You ask if a very good programmer can be a bad mathematician? I'd say it depends on what you mean by a good programmer.. you could make a fortune as a programmer who doesn't know lots of maths (isn't mathematician quality, anyway), yet I think those that win programming competitions and the likes are definitely mathematically inclined. I think just about all mathematicians can pick up programming quite easily, atleast if they had the interest to do so.

[quote="funcia"][quote="7h3.D3m0n.117"]Of course, my friend who is extremely profficient in math, doesn't even know what HTML is. Our curriculm suggests that we must learn HTML in seventh grade and nearly all of them have passed without a problem (even the ones in academic math).[/quote]
but i bet if you told them about html they'd soon be a wiz at it[/quote]

I did and he said "Whatever, I'm too lazy to learn it  :lol: "

I wish I had learned HTML in middle/high school :(  Anyway, learning HTML is more like learning a foreign language than learning C++, Java, Python, Perl, etc.

On the main topic, I know many mathy people who don't know much about programming (myself, in particular :D ).  However, I also know some mathy people who are really good at programming.  In general, though, I believe people who are good at math seem to pick up programming pretty easily.  The two are also related.  It helps to know some math to program and vice versa.

Actually I had to learn BASIC (for your TI calculators) from scratch. I didn't have a guide or anything and I had to look at basic programs my friend made to get the general idea. Then I found a section on Programming in the TI-83+ Handbook  :mad:  :ewpu:

Anyway I had a question. The Quadratic Formula is rather tedious to right out the work, etc. (don't worry the next part is about programming). So therefore I designed a simple program that asks you the value of a,b, and c in the quadratic equation and computes both the positive and negative (when applicible) answers. However, my friend got in deep trouble the other day because he knew how to program BASIC. Because tests measure how much you understand the topic, shouldn't programs like mine (the quadratic forumla one) be allowed because you did use the forumla except you made it less tedious for you.  :lol:

So by your logic, what's to stop you obtaining the answers from a previous class and entering them into your calculator? After all, you just made it 'easier' to take your test.

Hmm? If I had the answers there would be no need for my calculator.

[quote="max_tm"]What The Demon is trying to say is the fact that he programmed it himself should be proof enough that he understands the concept. While that is true there's no real way of checking that it's your original program; for all the teacher knows I could have given you the program 5 minutes before the test (assuming you know... we have a connector or IR or something) and then you could ace it without ever knowing the content.[/quote]

Exactly thank you except for the fact that I can program it while she watches (and while she scrubbs all the memory from the calculator). It only took about 45 seconds  :lol:


I had been solving a system of equation and have this conjecture.

In the systems of equation m1[i]x[/i] + n1y = p1 and m2x + n2y = p2, if the difference of m1, n1 and p1 are the same with the difference of m2, n2 and p2, then x + y = 1. I have a proof, just wanna know if there are other solutions.

Im not really good at proof so I wanna verify my answer.

[quote="10000th User"]Hmm, the problem with your problem statement is the term "the difference of ...". Difference, from what I understand, is the subtraction of quantity 2 from quantity 1 like $q_1-q_2$ I would say "the difference of $q_1$ and $q_2$". A more accurate use of this term is that "the difference [b]between[/b] $q_1$ and $q_2$" means $|q_1-q_2|$, with an absolute sign. I think this kind of difficulty arises because of language usage (I am not good either as English is my second language :) ). If you want to say $m_1-m_2=n_1-n_2=p_1-p_2$, you can say "...the differences between $m_1$, $n_1$, and $p_1$, and $m_2$, $n_2$, and $p_2$, respectively, are all the same..." I don't know whether my phrasing is correct and good but that's how I'll say it :)

About proof: make sure you are not dividing by zero, that both equations are not exactly the same: what I mean is $m_1-m_2=n_1-n_2=p_1-p_2\ne0$. I know this is obvious but you know you cannot divide by zero :)[/quote]

what i mean is n1-m1=p1-n1 = n2-m2 = p2-n2

[quote="10000th User"][quote="guile"]n1-m1=p1-n1 = n2-m2 = p2-n2[/quote]You should be able to see that this is equivalent to $m_1-m_2=n_1-n_2=p_1-p_2$ ... From this, $(m_1-m_2)x+(n_1-n_2)y=(p_1-p_2)$ and since each of these differences are equal and nonzero, $x+y=1$.

More challenging is the following:

Suppose you have $n_1-m_1=p_1-n_1\ne n_2-m_2=p_2-n_2$. Note that it is almost exactly as what you wrote above except there is a $\ne$ in it. Assuming $p_1m_2\ne p_2m_1$, you can find a unique value for $x$ and $y$. Find this $(x,y)$. Have fun :).[/quote]

Thanks. I just made up the problem. Actually the real problem was just investigation of systems with the same differences as mentioned above and no proof was asked. Thanks anyway. Cheers! AOPS rocks!

[quote="10000th User"]But have you tried my challenge? :D[/quote]

Not yet, ill try though if i have time, then ill post it. Whew! That's quite hard for a beginner...


actually i found it difficult to post the problem so i am adding an attachment.i want the answer to Q3.

well dat solution is beyond me.there ought to be a much easier one.

no one has got an easier solution.??the fact is that this is an RMO problem the beginning level for olympiads in india,so i thought there must some easy solution.


Given f(x) = x^3 - 6x^2 + 7x + 12, what is the sum of the squares of its zeros?

[quote="sonny"]We can write:

$ (x\minus{}n_{1})(x\minus{}n_{2})(x\minus{}n_{3}) \equal{} x^{3}\minus{}6x^{2}\plus{}7x\plus{}12$

$ (n_{3}\plus{}n_{2}\plus{}n_{1})^{2}\equal{} 6^{2}$

[/quote]

Where does the 6 come from?


Let $ f$ be a function for which $ f(x/3) \equal{} x^2 \plus{} x \plus{} 1$. Find the sum of all values of $ z$ for which $ f(3z) \equal{} 7$.

$ \textbf{(A)}\ \minus{} 1/3 \qquad \textbf{(B)}\ \minus{} 1/9 \qquad \textbf{(C)}\ 0 \qquad \textbf{(D)}\ 5/9 \qquad \textbf{(E)}\ 5/3$

Plug in $ x\equal{}9z$
$ f(\frac{9z}{3})\equal{}f(3z)\equal{}81z^2\plus{}9z\plus{}1$
$ 81z^2\plus{}9z\plus{}1\equal{}3 \rightarrow 81z^2\plus{}9z\minus{}2\equal{}0$
Sum of the roots is $ \frac {\minus{}b}{a}\equal{}\frac{\minus{}9}{81}\equal{}\frac{\minus{}1}{9}$
Answers is $ \boxed {B}$

[b]Inconsequential correction:[/b]
$ f(3x) \equal{} 7 \implies 81 x^2 \plus{} 9x \minus{} 6 \equal{} 0$, the answer is still the same though.  :)

Why did he set 81z^2 + 9z + 1 to equal 3? Shouldn't it be set equal to 7?

Oh, I thought he was posting a different method. Thanks for the clarification.


Prove that if a,b,c>0 , then

 $ \sqrt{\frac{a}{2a\plus{}b}}\plus{}\sqrt{\frac{b}{2b\plus{}c}}\plus{}\sqrt{\frac{c}{2c\plus{}a}}\geq \frac{\sqrt{3} (ab\plus{}bc\plus{}ca)}{a^2\plus{}b^2\plus{}c^2}$

My solution:
It suffices to prove that \[ \sqrt {\frac{a}{{2a \plus{} b}}}  \plus{} \sqrt {\frac{b}{{2b \plus{} c}}}  \plus{} \sqrt {\frac{c}{{2c \plus{} a}}}  \ge \frac{{(a \plus{} b \plus{} c)^2 }}{{\sqrt 3 (a^2  \plus{} b^2  \plus{} c^2 )}}\]
or \[ \frac{{a^2 }}{{a\sqrt {2a^2  \plus{} ab} }} \plus{} \frac{{b^2 }}{{b\sqrt {2b^2  \plus{} bc} }} \plus{} \frac{{c^2 }}{{c\sqrt {2c^2  \plus{} ca} }} \ge \frac{{(a \plus{} b \plus{} c)^2 }}{{\sqrt 3 (a^2  \plus{} b^2  \plus{} c^2 )}}\]
Using the Cauchy - Schuwarz ine, we have LSH\[ \ge \frac{{(a \plus{} b \plus{} c)^2 }}{{a\sqrt {2a^2  \plus{} ab}  \plus{} b\sqrt {2b^2  \plus{} bc}  \plus{} c\sqrt {2c^2  \plus{} ca} }}\]
But \[ a\sqrt {2a^2  \plus{} ab}  \plus{} b\sqrt {2b^2  \plus{} bc}  \plus{} c\sqrt {2c^2  \plus{} ca}  \le \sqrt 3 (a^2  \plus{} b^2  \plus{} c^2 )\]
So the ine is proved


geometry
geometric transformation
reflection

A convex quadrilateral has one vertex at $ (2,\minus{}1)$. The other three vertices are at the reflections of $ (2,\minus{}1)$ about the $ x$-axis, the $ y$-axis, and the line $ y\equal{}x$, respectively. What is the area of this quadrilateral?

By the given conditions, the points are $ (2,\minus{}1),(2,1),(\minus{}2,\minus{}1),(\minus{}1,2)$. Now, divide this into two triangles by drawing the diagonal connecting $ (\minus{}1,2)$ and $ (2,\minus{}1)$. The sum of these triangles' areas can easily be computed as (using lines parallel to the axes as bases) $ \frac12\cdot(2\minus{}(\minus{}2))\cdot(2\minus{}(\minus{}1))\plus{}\frac12\cdot(1\minus{}(\minus{}1))\cdot(2\minus{}(\minus{}1))\equal{}\boxed9$.


$a,b,c$ [size=150]are integers and atleast one of them is non-zero. Prove that:[/size]
${\left|\sqrt[3]{4}a+\sqrt[3]{2}b+c\right|\geq\frac{1}{4a^{2}+3b^{2}+2c^{2}}}$

well i posted this $ 2$ years ago... just wanted to remind that: "no one wants to solve this beautiful problem?!!!" :D


Here is a fun game. Lets teach a five year old math! Any suggestions on how? I have tried, but I don't know how to explain it. I want my little sister to be a math nerd like me, but its kind of hard when she still does not know addition or subtraction. :huh:

Read the thread "The cutest thing ever" for suggestions.

No, I'm kidding.


[quote="Differ"]If $ \sin\theta \plus{} \cos\theta \plus{} \tan\theta \plus{} \cot\theta \plus{} \sec\theta \plus{} \csc\theta \equal{} 7$, then find $ \sin2\theta$.[/quote]

$ (siin\theta\plus{}cos\theta)^{2}\equal{}1\plus{}sin2\theta$----> $ sin2\theta\equal{}22\minus{}8\sqrt{7}$

I don't quite understand your solution...
How did you get $ 22\minus{}8\sqrt{7}$?

[quote="Bugi"]I don't quite understand your solution...
How did you get $ 22 \minus{} 8\sqrt {7}$?[/quote]

$ (siin\theta\plus{}cos\theta)^{2}\equal{}1\plus{}sin2\theta$ and $ sin2\theta\equal{}a$

$ \sin\theta\plus{}\cos\theta\plus{}\tan\theta\plus{}\cot\theta\plus{}\sec\theta\plus{}\csc\theta\equal{}\sin\theta\plus{}\cos\theta\plus{}\frac{sin\theta}{cos\theta}\plus{}\frac{cos\theta}{sin\theta}\plus{}\frac{1}{sin\theta}\plus{}\frac{1}{cos\theta}\equal{}(\sin\theta\plus{}\cos\theta)(1\plus{}\frac{1}{sin\theta cos\theta})\plus{}\frac{1}{sin\theta cos\theta}\equal{}7$---> $ (\sin\theta\plus{}\cos\theta)^{2}\equal{}(\frac{7sin\theta cos\theta\minus{}1}{sin\theta cos\theta\plus{}1})^{2}$

$ 1\plus{}a\equal{}(\frac{7*\frac{a}{2}\minus{}1}{\frac{a}{2}\plus{}1})^{2}$----> $ a(a^{2}\minus{}44a\plus{}36)\equal{}0$----> $ a\equal{}22\minus{}8\sqrt{7}$


ratio
geometry
circumcircle
geometric transformation
homothety
geometry proposed

$ \triangle ABC$=isosceles ($ AB \equal{} BC$).
$ D$=midpoint of $ AB$.
$ O$=circumcenter of $ \triangle ABC$.
$ BE\bot CA$. $ F$=$ BE\cap$ Circumcircle (of $ \triangle BCD$).
Prove that $ 2BF \equal{} 3BO$.

May I ask you a favor ?  :) Thank you and Happy New Year !
I hope that all your resolutions will come true this year !

[b]REMARK:[/b] $ \mbox{The point F is circumcenter of }\triangle{ACD}.$

Yes, very easy ! Thank you for interest of my post.
Sorry for too easy question !
 :blush:

$ \angle FDC \equal{} \angle FBC \equal{} \angle EBA \equal{} \angle DCF$ so F stay on the perpendicolar axis of DC, i.e. F is the circumcenter of DAC, so F stay also on the perpendicolar axis of DA. Then call T the mispoint of DA, then BDO go in BTF with an homothety with factor $ \frac{2}{3}$ and center B.

[b]OK, ¬[ƒ(Gabriel)³²¹º]¼. [/b]
Simply...
[b]1).[/b] $ \angle {DBF}\equiv \angle{FBC}\Rightarrow \frown{DF}\equiv\frown{FC}\Rightarrow |DF| \equal{} |FC| \equal{} |FA|.$
[b]2).[/b] $ \mbox{If } T \mbox{midpoint of } [AD]\Rightarrow FT\perp AB \mbox{ and } OD\perp AB\Rightarrow$
$ \Rightarrow FT\parallel{}FT\Rightarrow \frac {|BF|}{|BO|} \equal{} \frac {|BT|}{|BD|} \equal{} \frac {3}{2}.$


geometry
rectangle
trigonometry
analytic geometry
graphing lines
slope
function

[asy]unitsize(50mm);
pair x2 = dir(50);
pair x1 = dir(25);
path p = (0,0)--(1,0)..x1..x2--cycle;
path q = (.361,0)--(.361,.434)--(.895,.434)--(.895,.0)--cycle;
draw(p);
draw(q);
label("$r$",(.355,.434),N);
label("$\theta$",(.1,0),N);[/asy]

1.) Find the side length of a square, which has one side along an edge of the sector, that can be inscribed in a sector with central angle $ \theta$ and radius $ r$, where $ 0<\theta\leq 90$.

2.) Determine, with proof, whether a square is the largest possible rectangle (in terms of area) that can be inscribed in such a manner.

[hide="Number 1"]
Blah, messed up, like always...
[/hide]

well, here's 1, at least

[hide]
Position the slice on a coordinate plane so that the side of the slice where the side of the square is on, is located right on the y axis. Now we see a configuration of a circle and a line intersecting. Let $ a$ be the the x coordinate in which they intersect and also the length of square. And let the slope $ b \equal{} tan(90 \minus{} \theta) \equal{} \frac {1}{tan(\theta)}$.

$ a \equal{} \sqrt {r^2 \minus{} a^2} \minus{} ba \Rightarrow a^2(b^2 \plus{} 2b \plus{} 2) \equal{} r^2 \Rightarrow a \equal{} \boxed{\sqrt {\frac {r^2}{(\frac {1}{tan(\theta)})^2 \plus{} \frac {2}{tan(\theta)} \plus{} 2}}}$[/hide]

[quote="Teki-Teki"][hide="#1"]
Let the side length of the square be $ x$.  Call the center of the sector $ O$, and let the endpoints of its arc be $ A$ and $ B$, such that the side of the square rests on $ OA$.  Let the square be $ KLMN$, with $ K,N$ on $ OA$, $ L$ on $ OB$, and $ M$ on $ \widehat{AB}$.

We have $ \frac {KL}{OK} = \cos{\theta},$ so $ OK = x\sec{\theta}$. Then $ ON = x(1 + \sec{\theta})$.  Also, $ MN = x$.  But $ OM = r$.  The Pythagorean theorem gives us $ ON^2 + MN^2 = OM^2$, so $ r^2 = x^2 + x^2(1 + \sec{\theta})^2$.  Hence $ x^2 = \frac {r^2}{2 + 2\sec{\theta} + \sec^2{\theta}}$, so $ x$ is just the square root of that junk.
[/hide]

[hide="#2"]
Define $ O,A,B,K,L,M,N$ as before, except using a rectangle instead of a square.

Let $ KL = MN = y$ and $ KN = LM = x$ (x and y same as the coordinate plane, using saillsha's image).  Then $ OK = y\sec{\theta}$.  Hence we have, as before, $ r^2 = x^2 + (x + y\sec{\theta})^2 = 2x^2 + 2xy\sec{\theta} + y^2\sec^2{\theta}$.  Make the substitution $ z = y\sec{\theta}$ and we find that $ r^2 = 2x^2 + 2xz + z^2$, which I don't feel like maximizing now.  There's probably a nice way to do it with Cauchy, but right now I don't see it and don't want to bother with Lagrange multipliers.
[/hide]

Edits: Fixed LaTeXing errors.[/quote]

For #1, $ \frac {KL}{OK} = \tan \theta$, not $ \cos \theta$... :huh:
So that would make $ OK = x \cot \theta$, and $ ON = x(1 + \cot \theta)$. This means that every occurrence of $ \sec$ must be replaced with $ \cot$, which gives the answer

$ x^2 = \frac {r^2}{2 + 2\cot \theta + \cot^2 \theta}$, so

$ x=\frac r{\sqrt {2 + 2\cot \theta + \cot^2 \theta}}$

Here's my solution for this one(question b is solved with TI-83 by graphing):
Let the center of that arc is O and A is the point lying on this arc. Draw a segment from O to A, we get: OA=r(const.)
Now, let the vertex of the square lies on the line marked r in the figure above be B, we have: OB=a/sin \theta
Applying the Laws of cosine in the triangle OAB we will have: r^2=AB^2+OB^2-2.AB.OB.cos(OAB)=a^2+a^2/sin^2 \theta+2a^2.cos \theta/(sin \theta)
Therefore, a= \frac r{\sqrt {2 + 2\cot \theta + \cot^2 \theta}}
From here, we see that to maximize the area of this square, we need to minimize {\sqrt {2 + 2\cot \theta + \cot^2 \theta}} where \theta is in the interval 0=<theta=<90.We write it in this form: (\cot \theta +1)^2+1.Since \cot \theta>0, we need to see the minimum of the function: f(theta)=\cot \theta in the interval 0=<theta=<90. Using TI-83, we will see that it attains the minimum value when theta=89.999979.Replacing it into the formula of a, we get the answer!(Q.E.D) :)
PS: Pleased check my solution since this problem is very "nice", especially part b) :P .Don't be afraid because of my lack of using LATEX cause I don't know how to use it!

Trivial
[hide]Call the center $ O$, and the four vertices of the rectangle be $ A,B,C,D$ clockwise from top-left. Let $ AO = x$, $ AB = y$, then $ AD = x\cdot \sin \theta$, and $ S_{ABCD} = xy\cdot \sin\theta$.
Now, applying Law of Cosines in $ \triangle AOB$,
\begin{align*} x^2 + y^2 + 2xy\cdot \cos \theta & = r^2 \\
2xy + (x - y)^2 + 2xy\cdot \cos \theta & = r^2 \\
2xy(1 + \cos \theta) & = r^2 - (x - y)^2 \end{align*}
Thus $ xy$ is maximized iff $ x = y$, or $ AB = x$. Hence, $ S_{ABCD}$ is maximized iff $ \boxed{AB: AD = 1: \sin \theta}$.
[/hide]

[b]@ghjk[/b]: Read the problem carefully. It asks for maximizing the rectangle; the angle is actually a constant.

Oh, I see! :blush:. But I think I already made a new problem by mistake throughout my solution. And it seems to be much harder when we are asked to find the maximum area of a square inscribed in an arc with a given length and an arbitrary angle in some intervals(like above is 0<=theta=<90),isn't it? :wink: .Can you try solving this problem, timwu?

[quote="ghjk"]Here's my solution for this one(question b is solved with TI-83 by graphing):

Let the center of that arc is O and A is the point lying on this arc.

Draw a segment from O to A, we get: $ OA = r$(const.)

Now, let the vertex of the square lies on the line marked r in the figure above be B, we have:
$ OB = a\sin \theta$

Applying the Laws of cosine in the triangle OAB we will have:

$ r^2 = AB^2 + OB^2 - 2\times AB\times OB\times \cos(OAB) = a^2 + a^2\sin^2 \theta + 2a^2\times\cos\theta(\sin \theta)$

Therefore,$ a = \frac {r}{\sqrt {2 + 2\cot \theta + \cot^2 \theta}}$

From here, we see that to maximize the area of this square, we need to minimize
$ {\sqrt {2 + 2\cot \theta + \cot^2 \theta}}$

where \theta is in the interval 0=<theta=<90.We write it in this form:
$ (\cot \theta + 1)^2 + 1$

Since \cot \theta>0, we need to see the minimum of the function: $ f(theta) = \cot \theta$ in the interval 0=<theta=<90.

Using TI-83, we will see that it attains the minimum value when theta=89.999979.Replacing it into the formula of a, we get the answer!(Q.E.D) :)

PS: Pleased check my solution since this problem is very "nice", especially part b) :P .Don't be afraid because of my lack of using LATEX cause I don't know how to use it![/quote]

Answer to your proposed problem:

The square yields maximal area.

You obtain the biggest square by letting $ \theta = \frac{\pi}{2}$

The area is $ \frac{r^2}{2}$

[quote="saillsha"] [color=darkred][b]1. [/b] Find the side length of a square, which has one side along an edge of the sector,
that can be inscribed in a sector with central angle $ \theta$ and radius $ r$, where $ 0 < \theta\leq 90$.

[b]2.[/b]  Determine whether a square is the largest possible rectangle (area) that can be inscribed in such a manner.[/color] [/quote]

[color=darkblue][b][u]Proof.[/u][/b] Call the center of the sector $ O$ and let the endpoints of its arc be $ A$ and $ B$ , such that the side of the rectangle rests on $ OA$ .

Let the rectangle  be $ KLMN$ with $ K,N$ on $ OA$ , $ L$ on $ OB$ and $ M$ on $ \widehat{AB}$ . Suppose w.l.o.g. that $ r \equal{} 1$ . Denote $ m(\widehat {AOM}) \equal{} x$ .

Then $ KL \equal{} MN \equal{} \sin x$ , $ KN \equal{} LM \equal{} \cos x \minus{} \sin x\cot \theta$ and the area of the rectangle is $ \sigma (x) \equal{} \sin x(\cos x \minus{} \cot\theta\cdot\sin x)$ .

Prove easily that $ \sigma'(x) \equal{} \cos 2x \minus{} \cot\theta\cdot\sin 2x$ . Observe that $ \left[\sigma '(x) \minus{} \sigma '\left(\frac {\theta}{2}\right)\right]$ and  $ \ \left(\frac {\theta}{2} \minus{} x\right)$ have same sign for any $ x\in\left(0,\frac {\pi}{2}\right)$ .

Therefore, $ \sigma(x)\le\sigma\left(\frac {\theta}{2}\right)$ for any $ x\in\left(0,\frac {\pi}{2}\right)$ and [b]maximum value is [/b]$ \boxed {\ \sigma\left(\frac {\theta}{2}\right) \equal{} \frac 12\tan\frac {\theta}{2}\ }$ .

[b]Remark.[/b] The rectangle $ KLMN$ is a square iff $ KN \equal{} MN$ $ \Longleftrightarrow$ $ \cos x \minus{} \cot\theta\cdot\sin x \equal{} \sin x$ $ \Longleftrightarrow$ $ \tan x \equal{} \frac {\tan \theta}{1 \plus{} \tan\theta}$

and in this case $ KL \equal{} \sin x \equal{} \frac {\tan x}{\sqrt {1 \plus{} \tan^2x}}$ , i.e. $ KL \equal{} \frac {\tan\theta}{\sqrt {1 \plus{} 2\tan\theta \plus{} 2\tan^2\theta}}$ .[/color]

Remark: Construction of the square:

As before, label $A, B$ endpoints of the arc and $O$ the center of the circle. Draw the altitude $BC (C\in OA)$, next draw the square $BCDE, D\in (CA$, $OE$ intersects the arc $\stackrel\frown {AB}$ at $M$; further, from $M$ draw parallels to $BC, OA$ respectively, which intersect $OA$ at $Q$, $OB$ at $N$, and lastly draw perpendicualr from $N$ to $OA$, call $P$ its foot. $MNPQ$ is the required square (its side is $MN=\frac{OM}{OE}\cdot BC$, and $OM=r$).

Best regards,
sunken rock


geometry
3D geometry
number theory proposed
number theory

Show that any integer can be expresses as a sum of two squares and a cube

This problem make me remember to Lagrange theorem.
"All non-negative number can express as the sum of four squares".

blang: how did you get those results ? Did you use computer or maybe some sophisticated method?  :D

A complete solution is given in The American
Mathematical Monthly, Volume 104, Number 6, June-July 1997, page 574. ;)


geometry
circumcircle
power of a point
radical axis
Pythagorean Theorem
geometry unsolved

Suppose that k is a circle with a diameter AB. Consider a random point C on k and the projection D of C to AB. The circle (C,CD) meets k at E,F. Prove that ,if EF meets CD at L and S is the projection of the midmoint of AD to EF, LC=LD=LS.  :!:

:blush:  Well, it's obvious that I should study INVERSION :oops:  :blush: . Anywhay, I have found a solution ( :idea: ) as follows: Firstly,we should prove that $ LC\equal{}LD$. To achieve that we extend $ CD$ until it meets again the two circles and then use the power of the point $ L$ with respect to the to circles. If $ DG$ is the altitude of the triangle $ CAD$,we have: $ GC*GA$=$ GD^2$=$ DC^2$-$ GC^2$. So $ G$ belongs to the radical axis $ EF$ of the two circles. The perpedincular from $ G$ to $ EF$ meets $ AB$ at $ R$. We have: $ <LRD$=$ <LGD$=$ <LDG$=$ <CAD$. So $ LR$ is parallel to $ AC$ and as $ L$ is the midpoint of $ CD$, $ R$ is the midpoint of $ AD$. So the point $ S$ is the point $ R$ and the result follows.

Let $M, N$ be the midpoints of $AB$ and $AD$, respectively. Note that $MC$ is the radical axis of the circle with diameter $AB$ and circle centered at $C$ with radius $CD$, so $MC\perp{EF}$. It follows that $NS\parallel{MC}$, so $\triangle{ANS}\sim\triangle{AMC}\iff\frac{AS}{AC}=\frac{AN}{AM}=\frac{AD}{AB}$, which gives $\triangle{ASD}\sim\triangle{ACB}\iff SD\parallel{BC}\iff \angle{CSD}=90^\circ$.

It suffices to show that $L$ is the midpoint of $CD$. By Power of the Point, we have $LE\cdot{LF}=CD^2-CL^2, LE\cdot{LF}=MC^2-ML^2$. It follows that, by Pythagorean Theorem, $CD^2-CL^2=MC^2-ML^2\iff CL^2-ML^2=MC^2-CD^2=MD^2$, which gives $MD^2=CL^2-(MD^2+LD^2)\iff LD^2=CL^2\iff CL=LD$.

Therefore, $CL=LD$, which implies that $L$ is the circumcenter of $CSD$ (since $\angle{CSD}=90^\circ$) and $LC=LD=LS$.

Inversion of pole $C$ and power $CD^2=CE^2=CF^2$ kills the problem!

Best regards,
sunken rock


Let $ABC$ be an equilateral triangle and let $P_0$ be a point outside this triangle, such that $\triangle{AP_0C}$ is an isoscele triangle with a right angle at $P_0$. A grasshopper starts from $P_0$ and turns around the triangle as follows. From $P_0$ the grasshopper jumps to $P_1$, which is the symmetric point of $P_0$ with respect to $A$. From $P_1$, the grasshopper jumps to $P_2$, which is the symmetric point of $P_1$ with respect to $B$. Then the grasshopper jumps to $P_3$ which is the symmetric point of $P_2$ with respect to $C$, and so on. Compare the distance $P_0P_1$ and $P_0P_n$. $n \in N$.

Let a = AB be the side of the equilateral triangle $\triangle ABC$. Since $AP_0 = AP_1,\ BP_1 = BP_2$, AB is a midline of the triangle $\triangle P_0P_1P_2$. Hence, the point $P_2$ is obtained from $P_0$ by a translation in the direction $\vec{AB}$ by the distance 2a. Similarly, $P_4$ is obtained from $P_2$ by a translation in the direction $\vec{CA}$ by the distance 2a and $P_6$ from $P_4$  by a translation  in the direction $\vec{BC}$ by the distance 2a. It follows that the points $P_0, P_2, P_4$ form an equilateral triangle with the side lengths 2a and the sides $P_0P_2 \parallel AB,\ P_2P_4 \parallel CA, P_4P_0 \parallel BC$ and that the point $P_6 \equiv P_0$ are identical. For similar reasons, $P_3$ is obtained from $P_1$ by a translation in the direction $\vec{BC}$ by the distance 2a, $P_5$ is obtained from $P_3$ by a translation in the direction $\vec{AB}$ by the distance 2a, and $P_7$ is obtained from $P_5$ by a translation in the direction $\vec{CA}$ by the distance 2a. It follows that the points $P_1, P_3, P_5$ also form an equilateral triangle with the side lengths 2a and the sides $P_1P_3 \parallel BC,\ P_3P_5 \parallel AB, P_5P_1 \parallel CA$ and that the point $P_7 \equiv P_1$ are identical.  Thus the equilateral triangles $\triangle P_0P_2P_4 \cong \triangle P_3P_5P_1$ are congruent and have parallel sides and the second is obtained from the first by a translation in the direction $\vec{P_0P_3}$ by the distance $P_0P_3$.

Up to now, we did not need the fact that the triangle $\triangle ABC$ is equilateral or that the triangle $\triangle AP_0C$ is isosceles, only the fact that $\vec{AB} + \vec{BC} + \vec{CA} = 0$, which holds for any triangle. Now we use these facts. Since $P_6 \equiv P_0$ are identical and $P_0A = P_0C, AP_0 = AP_1, CP_0 = CP_5$, the triangle $\triangle P_1P_0P_5$ is also isosceles. This means that the point $P_0$ lies on the altitude of the equilateral triangle $\triangle P_3P_5P_1$ from the vertex $P_3$. Since in addition, the isosceles triangle $\triangle P_1P_0P_5$ is right,

$P_0P_3 = P_1P_5 \dfrac{\sqrt 3}{2} - \dfrac{P_1P_5}{5} = a (\sqrt 3 - 1)$

Obviously, $P_0P_5 = P_0P_1 = 2AP_0 = 2CA \dfrac{\sqrt 2}{2} = a \sqrt 2$ and $P_0P_2 = P_0P_4 = 2a$. As a result,

$P_0P_2 = P_0P_4 = P_0P_1 \sqrt 2$

$P_0P_3 = P_0P_1 \dfrac{\sqrt 3 - 1}{\sqrt 2} = P_0P_1 \dfrac{\sqrt 6 - \sqrt 2}{2}$

$P_0P_5 = P_0P_1$

$P_0P_6 = 0$

$P_0P_n = P_0P_{(n \mod 6)}$


Could anyone tell me some good math books available on the net for free?especiall about geometry and combinatorics.

Find college's course websites. Some of the professors upload the textbooks (which is made by themselves)


As I am Indian myself, I wanted to start a discussion thread on the attacks a couple of days ago.

I heard that the USA FBI from LA is traveling there with a bomb squad and forensics people.  But it was pretty bad, except no crazy terrorists crashing into big towers.

Even though the Prime Minister didn't point any fingers and merely said "the attackers are foreigners", I am scared about how this event will affect the fragile unity between Hindus and Muslims. On the other hand, I doubt that the terrorists will accomplish their goal of permanently disabling India's international business.  :)

Just now I got the news five more hostages are killed.
Things are getting worse.
I don't have words to describe how cruelly they have killed these innocent people.
All the three hotels had turned into house of death bodies.
I don't care whether the NSG kill those terrorists or arrest them.
But what I want at this moment is no more killing of innocent people.
May Jesus give strength to those family who have lost there loving ones in this attack.

I remembered one of the favorite poem of Gandhi Ji which we used to sing in school
Isswar  Allah tere jahaa mein nafrat kuon hain
jang hain kuon
Tera dil to itna bada hain
Inshaan ka dil tang hain kuon hain

Although more intricately planned and with more innocent lives at stake, this event seems similar to the 1980 Iranian Embassy Siege. I hope this will end as soon as possible so officials can figure out exactly why the attacks were carried out and what group is responsible for this.

wait u're Indian ernie?  :o

and yeah its pretty bad over there...
poor them

It's pretty ironic, because the security was loosened a couple of days ago.

And India has verified that the terrorists were Pakistani and some of them were from Europe (possibly those who participated in the London bombings and the LET).

Remember that this all started up again on July 25th (ironically, my birthday), with the bombings in Bangalore.  The next day, there were bombings in my hometown city, Ahmadabad.  Since then, there were 2 more bombings in Delhi and Assam.  But India was able to pick itself up and move on.  However, this attack was much more terrible, killing over 160 people and injuring more than 350 more.

Since then, we have taken back the Oberoi and the Nariman house.  But we have yet to recover the once-beautiful Taj hotel.

[size=200]I know just the thing to do to those terrorists: Cut off their body parts one by one before killing them. Also remove the organs. That should ensure maximum pain. And this execution should be on camera, so any other terrorists with bad plans know their fate. It serves them right for what they did to India. Maybe we will not kill them, we will let those [/size][size=9]jackasses[/size][size=200] live in pain and suffering. That should put an end to terrorism.[/size]

[quote="jain_harshi"]You are right. I lived very close to Mumbai and know about the Taj. Seeing it burning is just heartbreaking. I just wonder, why did the terrorists attack a hospital? Do you know they had taken patients hostage there, but released them?[/quote]

This tactic was used in the Ahmadabad bombings.  All the injured were admitted to a hospital, which was later bombed.

[quote="mathelete"]I know just the thing to do to those terrorists: Cut off their body parts one by one before killing them. Also remove the organs. That should ensure maximum pain. And this execution should be on camera, so any other terrorists with bad plans know their fate. It serves them right for what they did to India. Maybe we will not kill them, we will let those jackasses live in pain and suffering. That should put an end to terrorism.[/quote]

No it won't.  But we need to take all the information out of them too.

[quote="mathelete"]I know just the thing to do to those terrorists: Cut off their body parts one by one before killing them. Also remove the organs. That should ensure maximum pain. And this execution should be on camera, so any other terrorists with bad plans know their fate. It serves them right for what they did to India. Maybe we will not kill them, we will let those [/size][size=9]jackasses[/size] live in pain and suffering. That should put an end to terrorism.[/size][/quote]
Holy crap. That's scary.
And... no. That would only inspire more violence. As much as some Americans (reminded of Palin here) see the terrorists as evil, they are only fighting for their beliefs. If you think about it, our Founding Fathers were terrorists too...
And taking all the information out of them also sounds scary. "Enhanced interrogation techniques," anyone?

[quote="Brut3Forc3"]And taking all the information out of them also sounds scary. "Enhanced interrogation techniques," anyone?[/quote]
Do you have any idea what [b]their[/b] techniques did to us and more specifically to how many of us and to how much things of ours?
They knew they would die when they started all these. So we need to do something more with them so that no one dares it again.

And yes the Pakistan refused to send ISI chief to India:
http://timesofindia.indiatimes.com/World/Pak_to_send_official_not_ISI_chief/articleshow/3771372.cms

The started violence and cruelity, so you are allowed to use it too¿ That's a rather dangerous and abandoned view, it just causes hate and an infinite circle of even more hate.
And after your "very reasonable interrogation techniques" were used against them, the next step will be to use them against people you aren't even sure about their "terroristic activities". But after enough "interrogation", they surely will confess it.. You know, european witch hunting was very successfull, most of them confessed to be witches... (and the those who didn't confess were obviously guilty for collecting herbs or having birthmarks)
You didn't understand some implications of human rights; and mathlete's postis one of the stupidiest posts I've ever seen on this forum. He may be angry at them, but this is no reason to talk about cruel murders of persons (even if terrorists). In fact, mathlete is even worse than some terrorists, he puts judge on them and shows some execution video to the world; his post sounds more like some execution video of the taliban than words of an intelligent being that has considered a bit more than his useless revenge. Maybe you haven't thought of this, but you would act no better than them; so why shouldn't they in reverse do the same to you¿ What's the exact point in acting exactly like them¿ Showing them even more, that this actions are accepted on all sides¿ Showing them, that you murder their friends and relatives in a very cruel way, so that they hate you and your nation more than everything else, causing more attacks¿ Thinking further, you acting that way may even be their plan: it gives them more supporters (in two ways) and it attacks your society (like you) by alternating them into wishfull murderers.
And writing that post in bold and huge letters is just annoying, don't do this ever again.

[quote="mathelete"]I know just the thing to do to those terrorists: Cut off their body parts one by one before killing them. Also remove the organs. That should ensure maximum pain. And this execution should be on camera, so any other terrorists with bad plans know their fate. It serves them right for what they did to India. Maybe we will not kill them, we will let those [/size][size=9]jackasses[/size][size=200] live in pain and suffering. That should put an end to terrorism.[/quote]

mathelete
We all are sad.I personally know some people who have lost their loved ones.
That's not mean that we should behave like them .
I have seen the images of those terrorists .They would be less than 25 years.Can't u think why they are doing this when they know that they would surely be killed by soldiers and police.Remember in Delhi's blast the terrorist were students of Jamiya university .
The reason is they don't know what humanity is?They just became a destructive machine.
If we want to get rid of terror,we have to find the root of terror.
Once I was watching a video in which it was shown that how children nearly 10-15 years old are trained to become a terrorist.
If u really want to do something ,do something which can save these kids to become a terrorist.

At last I just wanna say that "In real life , two negative never make a positive".
Hopefully u would understand.

[quote="ZetaX"]The started violence and cruelity, so you are allowed to use it too¿ That's a rather dangerous and abandoned view, it just causes hate and an infinite circle of even more hate.[/quote]
hmm, true. But I don't really understand their point of view. I mean they knew they would die, didn't they? Yes there is no point in cruelty. But we can't let them loose. They should be imprisoned, right? the few who are alive?

[quote="ZetaX"]And after your "very reasonable interrogation techniques" were used against them, the next step will be to use them against people you aren't even sure about their "terroristic activities". But after enough "interrogation", they surely will confess it.. You know, european witch hunting was very successfull, most of them confessed to be witches... (and the those who didn't confess were obviously guilty for collecting herbs or having birthmarks)[/quote]
I never agree to "interrogating" ppl on the basis of suspecting. But things like who sent them and similar things should be asked. That's all I wanted to say.

[quote="ZetaX"]You didn't understand some implications of human rights; and mathlete's post is one of the stupidiest posts ... this ever again.[/quote]
I never agreed to mathlete's post. I only replied to the quote of the previous post.

P.S. Why are all your "?"s inverted?

[quote="Akashnil"]I mean they knew they would die, didn't they?[/quote]

These people were willing to die.

Let's see what happened:

About 15 terrorists hijacked 10 different places, including two luxury hotels (Taj and Oberoi), a Jewish center, and a hospital.  They just ran in with AK-47's, hand grenades, and started randomly opening fire.  They kept hostages, and they killed around 170 people, injuring around 350 more.

Wow, they even hijacked a police car and started shooting from there.

But I only got this from the media.

And see this: HT said that there were 15 terrorists.  Another news line (24-7?) said that there were 40.  You can't clearly trust the media very much.

[quote="Akashnil"][quote="ernie"]These people were willing to die.[/quote]
I know that. But my question is WHY?[/quote]

They are ready to give up their life, they hate the world, and want to destroy it.  And how would I know?

Probably they are brainwashed, similar to what happened in Nazi Germany, where children were brought up with an overwhelming deluge of pro-Nazi and anti-Jew propaganda. Terrorists can become fanatic in their beliefs from years of careful conditioning by their leaders to believe they are the saviors of the world. Even ordinary people can fall into this too.
Case in point: the Crusades. The Europeans and the Muslims, on their "war for religion", probably never stopped to look at their opponents. If they did, they would have realized they were fighting a people who believed in basically the exact same religion.
The media is especially good at doing this. People have a tendency to trust everything they hear, as long as it's not contradicted. When the terrorists were in their camps in the Pakistani mountains or wherever, they only heard pro-Muslim and anti-Indian messages, thus leading them to believe they were fighting or a just cause. I am not trying to defend their acts, just explain them. If we can understand why these things occur then it will be easier to stop future occurrences.

[quote="kabi"]
At last I just wanna say that "In real life, two negatives never make a positive".
Hopefully u would understand.[/quote]

Then what do we do, let them live so they can attack again?

[quote="mathelete"]

Then what do we do, let them live so they can attack again?[/quote]
Dear why don't u understand ?
There are in millions of people  who are in these type of ridiculous works.
O.K. fine if we all start doing what u have said then will they stop all these.They will some other.
They all have lost their soul.
All of them who are involved in this should be punished but not in the way in which u want.
If u really want to help India destroy the roots of terrorism .
These terrorists group take children from their houses and give them training .Why can't u understand that why they are doing this?
Don't waste ur energy in finding useless works .
Ur country need u .Do something which can really prevent this world from terrorism.
I meant to a place where terrorists had made their home but I don't want ur to become the same.I too have lost my near ones in an attack.So my responsibility to stop these all has  increased.
I  m really feeling sorry for u.

[quote="Akashnil"][quote="ernie"]These people were willing to die.[/quote]
I know that. But my question is WHY?[/quote]

Most terrorist groups are started by religious extremists who interpret their religious texts in a rather twisted/irrational way (the most common idea being that violence is justified to eradicate other religions, to put it crudely). However, these leaders obviously need a large force behind them, so they turn to regions of extreme poverty. Usually, they recruit young boys with the lure of money/food/etc for their families. Of course, these young men see no other choice (they'll be taken in by force if they don't agree anyway), so they join the training camp, where they gradually become brainwashed (as someone said). They are literally void of emotions; they're highly trained to kill, even if it means giving up their own life.

I guess I might be violating some of the "The Round Table" rules (reviving threads,etc), but I would like to pay homage to all the victims of this ghastly act  of terrorism-- perpetrated by Pakistani terrorists. Today is the first anniversary of 26/11 Mumbai terror attacks-- one of the most horrific instances of terrorism the world has ever witnessed.

On a discursive note, I would like to note that one can't compare this attack to "9/11" (as the [b] OP [/b] had done) primarily because the extent of damage, the number of casualties etc pales in comparison with the "9/11 attacks".

[b] May the souls of all victims rest in peace.[/b]

Don't worry!Such things will continue till Pakistan is there!And we Indians have got used to it!They have been pushing terrorists into India for the last 60 years!
It took years for people in the west to wake up to the reality of Pak.Nobody cares till one is affected.And Obama talks of nuclear disarmament.

I guess by this time (exactly an year ago!) the two terrorists had finished their massacre at the CST and were heading for Cama hospital. By this time, I guess several Americans and other foreigners were also taken hostage at the Taj. I once again pay homage to all the bereaved families of foreigners who were killed in this savage act of terrorism.

[quote="nobelium"]They have been pushing terrorists into India for the last 60 years! [/quote]

You're mistaken. They have been using terrorism as a state policy-- by starting insurgency in Kashmir from almost 1988-- the time by which they had or had developed the N-bomb. As soon as they had obtained the N-bomb-- the levels of insurgency in Kashmir started and increased dramatically over the years. Fomenting unrest in India was always one of their most important state policies. So, by having weapons of Mass Destruction in their arsenal-- they can always  resort to nuclear blackmail-- after committing grave and savage outrages like 26/11.

[quote="nobelium"] It took years for people in the west to wake up to the reality of Pak.Nobody cares till one is affected.And Obama talks of nuclear disarmament.[/quote]

Quite unfortunately, they're still sleeping! They're fighting the Al-Qaeda, but they're not realizing the grave threat posed by terrorist organizations like the Lashkar-e-Taiba (LeT) and others. They have to secure Pakistan's nuclear weapons (as quickly as possible)-- which pose an imminent threat to world peace and stability. Imagine a scenario in which a terrorist organization seizes these weapons.

They might not launch these weapons against the West, but the West should understand the fallout of a nuclear war and its effects on global climate. They should also understand the threat posed by nuclear proliferation by various rogues states or scientists. But, I opine that the U.S. should secure or wrest control of these weapons as soon as possible.

[b] P.S.: [/b] All of my views/opinions are known facts. I don't intend to offend anyone.

[quote="Gen8"]
You're mistaken. They have been using terrorism as a state policy-- by starting insurgency in Kashmir from almost 1988-- the time by which they had or had developed the N-bomb. As soon as they had obtained the N-bomb-- the levels of insurgency in Kashmir started and increased dramatically over the years. Fomenting unrest in India was always one of their most important state policies. So, by having weapons of Mass Destruction in their arsenal-- they can always  resort to nuclear blackmail-- after committing grave and savage outrages like 26/11.
[/quote]

They pushed the villainous pathans into India immediately after partition.They massacred thousands.It too was a form of terrorism.Pathans were not part of regular army of Pak.
By the way,which country do you live in?

[quote="nobelium"] They pushed the villainous pathans into India immediately after partition.They massacred thousands.It too was a form of terrorism. [/quote]

Okay, But we forgot that they instigated Sikhs too. (In the 80s) They fomented Sikh terrorism too. Also, I would concur with your assessment that these terror attacks will continue as long as Pakistan pursues its policy of sponsoring terrorism against its neighbours.

[quote="nobelium"]Anyway,let's stop it here.No use being so harsh on them.[/quote]

Well, I was only stating the facts. (They're all facts!)

I apologize for posting-- which contravenes [b] JBL's [/b] policies regarding the The Round Table.  I had just read 'this' sometime ago and it's making me shiver and shudder.

I will just post the gist of the 'news report' over here. A noted U.S. counter-terrorism official had said that if another attack akin to 26/11 were to occur in India, it could lead India and Pakistan to engage themselves in a nuclear war-- a hair-raising prospect indeed. (The newspaper is the link below is extremely well reputed in India; it ain't like the NY Post)

[url]http://timesofindia.indiatimes.com/india/India-may-strike-terror-training-camps-post-Mumbai-II-US-counter-terrorism-expert/articleshow/6570351.cms[/url]

Secondly, the fallout of a regional nuclear war would be devastating to the climate too.

[quote="evilhamster"]
Case in point: the Crusades. The Europeans and the Muslims, on their "war for religion", probably never stopped to look at their opponents. If they did, they would have realized they were fighting a people who believed in basically the exact same religion.[/quote]

L2crusades please...they were mainly to unite Europe politically and maintain the power of the Church. Europe was extremely unstable at the time and a shared goal helped to bring order to the numerous kingdoms.