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  #11  
Old 11-25-2007, 05:40 AM
Phil153 Phil153 is offline
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Default Re: Two Olympiad Problems

When are you going to write that book Mr Sklansky? You've lived a life of sin and debauchery and if there is a God he won't be pleased.

Maybe a brilliant textbook will get you over the line.
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  #12  
Old 11-25-2007, 06:47 AM
bigpooch bigpooch is offline
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Default Re: Two Olympiad Problems

The "elliptical idea" seems the most elegant IMO (but I'm
biased since this was the idea that struck immediately).
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  #13  
Old 11-25-2007, 11:23 AM
TomCowley TomCowley is offline
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Default Re: Two Olympiad Problems

[ QUOTE ]
Here is the way that requires neither geometrical ingenuity nor taking second derivatives of trigonemetric functions. (If Tom Cowley does that again his reign here will be short lived.) Take a string six inches long and attach the ends. Thumbtack a potion of it on a horizontal line.Pull the rest up to make a triangle and notice that it is highest, and thus has greatest area, when it is isosceles.

[/ QUOTE ]

Bah. That's exactly the observation I made, except I gave a proof that it was true instead of just stating that it was true.
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  #14  
Old 11-25-2007, 04:38 PM
pzhon pzhon is offline
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Default Re: Two Olympiad Problems

You don't have to use trigonometry to prove that the height is maximized when the triangle is isoceles. You can use the triangle inequality after adding a reflected copy of the triangle above. So, this whole problem can be done without calculus.
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  #15  
Old 11-25-2007, 06:03 PM
David Sklansky David Sklansky is offline
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Default Re: Two Olympiad Problems

[ QUOTE ]
You don't have to use trigonometry to prove that the height is maximized when the triangle is isoceles. You can use the triangle inequality after adding a reflected copy of the triangle above. So, this whole problem can be done without calculus.

[/ QUOTE ]

All calculus problems can be done without calculus. When my father was teaching mathematical logic at City College he would sometimes ask for a calculus problem and demonstrate.
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  #16  
Old 11-25-2007, 06:39 PM
gumpzilla gumpzilla is offline
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Default Re: Two Olympiad Problems

[ QUOTE ]
[ QUOTE ]
You don't have to use trigonometry to prove that the height is maximized when the triangle is isoceles. You can use the triangle inequality after adding a reflected copy of the triangle above. So, this whole problem can be done without calculus.

[/ QUOTE ]

All calculus problems can be done without calculus. When my father was teaching mathematical logic at City College he would sometimes ask for a calculus problem and demonstrate.

[/ QUOTE ]

Two comments:

1) I've never understood the resistance to using calculus on these types of problems, as if it is some kind of failure. Finding different roads to a solution is often very interesting, but what's wrong with calculus?

2) All calculus problems? How would one frame a generic differential equation in language that does not involve calculus? Is there some kind of geometric trickery that will give me Bessel functions?
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  #17  
Old 11-25-2007, 06:59 PM
pzhon pzhon is offline
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Default Re: Two Olympiad Problems

[ QUOTE ]
I've never understood the resistance to using calculus on these types of problems, as if it is some kind of failure. Finding different roads to a solution is often very interesting, but what's wrong with calculus?

[/ QUOTE ]
First, it makes the solution more opaque. I'm not so interested in solving any one problem. I'm interested in acquiring powerful techniques which will let me attack additional problems. Since I know calculus, seeing a calculus solution generally does not help me with future problems, but seeing something like my reflection argument very well might.

Second, the non-calculus arguments were much simpler and easier to remember, both to show A=B, and then A=B=C.

[ QUOTE ]
All calculus problems? How would one frame a generic differential equation in language that does not involve calculus? Is there some kind of geometric trickery that will give me Bessel functions?

[/ QUOTE ]
Right, I don't buy that. Parts of calculus may be formally removable, but only by greatly increasing the complexity of calculations in general. I don't want to see someone try to remove calculus from a simple optimization with Lagrange multipliers, or a curvature calculation, or a deconvolution to sharpen a blurry image.
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  #18  
Old 11-25-2007, 11:05 PM
borisp borisp is offline
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Default Re: Two Olympiad Problems

[ QUOTE ]
All calculus problems can be done without calculus.

[/ QUOTE ]
Yes, and all computer programs don't need computers to run them. I file this sort of thing under "true but irrelevant."

However, I do generally agree that a simple "stick in the sand" solution is preferred over one that mindlessly uses a technique. BUT, given that one understands the proof of why the technique works, the general method becomes more attractive. Lagrange multipliers (from another thread) is a good example, since the proof can be motivated very easily using geometry of graphs, tangent planes, etc. Of course, this is mostly speaking for advanced math students.

But this brings me to my question: DS, why are you so bent on explaining things in a way so that the average math student can understand them? I can understand the motivation to sell your publications, so if that is all, then so be it.

Beyond that, do you really think these simpler explanations really improve the average student's day to day decision making? If they weren't smart enough to solve a problem to begin with, what makes you think that they will be smart enough to recognize when they should be clever and actually employ what they have learned? I've found (in my own teaching) that the latter is really the most difficult hurdle. You can make people understand things, but you can't make them recognize when it is appropriate to apply what they have understood. That is, unless they are smart enough to understand more difficult and general techniques, and then we are back where we started.
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  #19  
Old 11-26-2007, 12:21 AM
David Sklansky David Sklansky is offline
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Default Re: Two Olympiad Problems

[ QUOTE ]
[ QUOTE ]
All calculus problems can be done without calculus.

[/ QUOTE ]
Yes, and all computer programs don't need computers to run them. I file this sort of thing under "true but irrelevant."

However, I do generally agree that a simple "stick in the sand" solution is preferred over one that mindlessly uses a technique. BUT, given that one understands the proof of why the technique works, the general method becomes more attractive. Lagrange multipliers (from another thread) is a good example, since the proof can be motivated very easily using geometry of graphs, tangent planes, etc. Of course, this is mostly speaking for advanced math students.

But this brings me to my question: DS, why are you so bent on explaining things in a way so that the average math student can understand them? I can understand the motivation to sell your publications, so if that is all, then so be it.

Beyond that, do you really think these simpler explanations really improve the average student's day to day decision making? If they weren't smart enough to solve a problem to begin with, what makes you think that they will be smart enough to recognize when they should be clever and actually employ what they have learned? I've found (in my own teaching) that the latter is really the most difficult hurdle. You can make people understand things, but you can't make them recognize when it is appropriate to apply what they have understood. That is, unless they are smart enough to understand more difficult and general techniques, and then we are back where we started.

[/ QUOTE ]

Its strange that you used this thread as impetus for this question. Because my answer was not the one that used the least math, though I do think it was the easiest to understand. If you know basic calculus.

Anyway it isn't people who aren't smart enough to understand more difficult and general techniques that I try to reach. Its only the twenty percent or so of the population who could, with difficulty, understand them, but choose not to learn them.
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