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I am not looking for contest problems where there is a clever trick or a standard approach, I am looking for more creative and open-ended problems such as this , and I am not looking for questions like this. Thanks.

(I have to admit the first problem is actually from a contest, but it feel very different from the other ones)

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Ovi
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  • Well, you could attack physics. Many problems there are in the form of "find out what happens in the following situation". Also, there is the nice fact that you can try and reproduce your results experimentally (at least in the not too advanced stuff). – Daniel Robert-Nicoud Aug 15 '13 at 22:34
  • @DanielRobert-Nicoud do you know a good online source of mechanics problems? I don't currently own a physics textbook – Ovi Aug 15 '13 at 22:36
  • I'm afraid I don't. If I find something I'll let you know, or maybe with a bit of luck someone else reading this question will have a source. – Daniel Robert-Nicoud Aug 15 '13 at 22:47
  • @DanielRobert-Nicoud Ok thanks – Ovi Aug 15 '13 at 22:50
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    I'm confused by this question. The problem you cited as "open-ended" is just a standard contest problem like the first problem you linked, and it has a fairly straightforward solution. Could you clarify what you're looking for? Do you just want harder (or Olympiad-style) contest problems? – Potato Aug 16 '13 at 08:14
  • @Potato I guess what I'm looking for is problems which don't rely mainly on heavy/tricky algebraic manipulation to make them hard. The Olympiad questions are good, but they are a bit too hard for me and many times I haven't learned the material required for a solution. – Ovi Aug 16 '13 at 13:49
  • @Ovi Take a look at Problem Solving Strategies by Engel. Is that more like what you want? – Potato Aug 16 '13 at 17:56
  • @Potato I have that book, but that is what I'm trying to avoid. It is a book that teaches a lesson/approach and then it gives a bunch of problems using that approach, and usually unless you use that approach the problem is extremely difficult to solve. But I am looking for more open ended problems, where you don't know the approach needed and there is more than one possible way to approach it. – Ovi Aug 16 '13 at 18:06
  • @Potato for example, the problem with the dartboard was solved with a functional equation, but I remember I discovered at least another way to do it. But most other contest problems (from my experience) there is only one predestined approach, and if you don't use that one the problem becomes extremely difficult to solve – Ovi Aug 16 '13 at 18:10
  • @Ovi There's really only one way to do the dartboard problem. You note that the area you need is bounded by $4$ parabolas, then you find the area (by integration, for example). You just re-derived the equation for a parabola. So I'm not sure what you mean, because there is really just one essential way to do that problem. (As far as I can see.) – Potato Aug 16 '13 at 18:16
  • @Ovi If you're concerned about having the approach spelled out for you, like in Engel's book, why not just take a look at old olympiads, like the USAMO? Or, since you said those are too hard, try the easier ones like the Canadian Math Olympiad and the Irish Math Olympiad. – Potato Aug 16 '13 at 18:18
  • @Potato What I meant is that I used trigonometry to find f(x) instead of using a functional equation. And thanks, I didn't know that the Canadian and Irish Olympiads are easier. But which would you say is the easiest? – Ovi Aug 16 '13 at 18:19
  • @Ovi I'm not sure, but some of the old Canadian tests are very approachable. (And if you're having trouble with them, focused practice with a book like Engel's is a great way to get better...) – Potato Aug 16 '13 at 18:23
  • @Potato Ok thanks – Ovi Aug 16 '13 at 18:25
  • @Ovi Maybe you’ll be interested in problems from “Math Magic” by Erich Friedman. This site mostly contains different combinatorial puzzle problems. They are very creative and open-ended, :-) their hardness is varied. I was solving these problems when I was younger. :-) Erich adds a new problem to the site every month. – Alex Ravsky Aug 18 '13 at 03:35

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