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ihor
Supreme Hero
Accidental Hero
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posted June 29, 2010 08:09 PM |
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You can.
For example as in my right picture you can find a point D on bottom line such that distances CD and AB are equal.
Though I guess you can dispense with it.
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friendofgunnar
Honorable
Legendary Hero
able to speed up time
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posted June 29, 2010 10:47 PM |
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Here's one (probably inefficient) solution.
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ihor
Supreme Hero
Accidental Hero
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posted June 30, 2010 08:42 AM |
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Edited by ihor at 08:56, 30 Jun 2010.
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Explain please your step 4.
It is not obvious how to build such a triangle and I afraid it is not correct 
BTW efficiency doesn't matter in this problem.
Edit:
Suitable joke:
Physicist and mathematician have the same problem - you have a kettle, tap and cooker, how to boil water in the kettle?
Physicist: turn on water tap, pour water in the kettle, put the kettle onto cooker, turn on cooker.
Mathematician: turn on water tap, pour water in the kettle, put the kettle onto cooker, turn on cooker.
Next problem for them - you have the same but there is already water in the kettle.
Physicist: put the kettle onto cooker, turn on cooker.
Mathematician: empty out the kettle and use the above techniques to do that.
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friendofgunnar
Honorable
Legendary Hero
able to speed up time
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posted June 30, 2010 10:35 AM |
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I was wondering if that part was clear enough...
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ihor
Supreme Hero
Accidental Hero
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posted June 30, 2010 12:14 PM |
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Edited by ihor at 12:15, 30 Jun 2010.
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Aha great!
I thought about another triangle when you said "small right" 
Thats correct.
And your solution is more efficient than mine.
I'm too lazy to present it all. Just generally:
1)Parallel line technique.
2)Halving line segment technique. (using Thales's theorem)
3)Building such a triangle in which we can easy build 2 altitudes(for example using halving line segment).
4)Building final third altitude, which is our desired line (using the fact 3 altitudes intersect in a single point).
So what about your circle and square (Q23)? Any hints?
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Ecoris
Promising
Supreme Hero
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posted June 30, 2010 12:26 PM |
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ihor
Supreme Hero
Accidental Hero
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posted July 01, 2010 02:33 PM |
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Ha ha I knew there is a catch in it.
Though history knows situations when open problems were put among other tasks on different olympiads, so they have been solved in this way.
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friendofgunnar
Honorable
Legendary Hero
able to speed up time
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posted July 03, 2010 05:14 PM |
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heheh, yeah Ecoris has revealed the answer. I was going to show my strategy for squaring a circle to an arbitrary degree of precision but I can know see that whatever I can think of has almost assuredly been thought of before.
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dimis
Responsible
Supreme Hero
Digitally signed by FoG
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posted July 04, 2010 01:15 AM |
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Edited by dimis at 01:27, 04 Jul 2010.
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AlexSpl
Responsible
Supreme Hero
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posted July 19, 2010 02:12 PM |
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Could you please help me with one question?
Imagine some closed room with broken electric bulb in it. The bulb switches on and off periodically. A man may come into this room from 10.00 to 11.00 but nobody can say when exactly he will come (he comes at random time between 10 and 11). Let M(t) is function that determines when the bulb is switched on and when switched off: M(t) = 1 for switched on state, 0 otherwise (t is an integer in seconds). What the man's chances to open the door to the room and see the bulb switched on? Are they Pr[On] = Sum{t = 1..3600 s} [M(t)] / 3600 s?
Let now M(t) is much more often takes 0 value than 1 (most probably the bulb is in off state). If the man came earlier, say at 10.05, what his chances to see the light when opening the door, compared to those if, instead, he would have came at 10.35? Note: M(t) is known function but only for those who is inside the room.
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friendofgunnar
Honorable
Legendary Hero
able to speed up time
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posted July 19, 2010 02:24 PM |
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Quote:
Why Dating Is Difficult in Big Cities
Because there is a proof.
This is why it's much easier to date in a french fry pit than it is in college. In a french fry pit you only have choices a,b,and c. In college you have choices [A1, A2,...A496,A497]
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Ecoris
Promising
Supreme Hero
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posted July 19, 2010 06:31 PM |
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Quote:
Imagine some closed room with broken electric bulb in it. The bulb switches on and off periodically. A man may come into this room from 10.00 to 11.00 but nobody can say when exactly he will come (he comes at random time between 10 and 11). Let M(t) is function that determines when the bulb is switched on and when switched off: M(t) = 1 for switched on state, 0 otherwise (t is an integer in seconds). What the man's chances to open the door to the room and see the bulb switched on? Are they Pr[On] = Sum{t = 1..3600 s} [M(t)] / 3600 s?
Yes, obviously. It's the integral.
Quote:
Let now M(t) is much more often takes 0 value than 1 (most probably the bulb is in off state). If the man came earlier, say at 10.05, what his chances to see the light when opening the door, compared to those if, instead, he would have came at 10.35? Note: M(t) is known function but only for those who is inside the room.
I don't understand your question. With no knowledge of the bulb's behaviour the probabilities are the same.
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AlexSpl
Responsible
Supreme Hero
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posted July 20, 2010 07:47 AM |
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Edited by AlexSpl at 07:49, 20 Jul 2010.
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Thank you, Ecoris.
Quote:
Note: M(t) is known function but only for those who is inside the room.
OK. Here is the problem. The man sitting inside the room knows for sure the answer: Pr[On @ 10.05] = M(5 * 60), Pr[On @ 10.35] = M(35 * 60), isn't he? Why should the man outside think that those probabilities are equal, because of the lack of knowledge (initial data), right? But this is so strange...
Suppose, M(t) is known function. Well, now, the man comes at random time between 10.00 and 10.05, and another man comes between 10.00 and 10.35. For which man probability to see the light is greater? (Remember: M(t) tends to be zero rather than 1).
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dimis
Responsible
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Digitally signed by FoG
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posted July 20, 2010 09:15 AM |
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Edited by dimis at 09:26, 20 Jul 2010.
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Ok, first things first.
I guess you associate this problem, with the discussion in the "Not in the Manual" thread that recently started. This wasn't mentioned, so, it is does not necessarily follow that others can follow your thoughts if you omit some details. For example, I never have an email-notification for any thread, so, it may happen that I don't see something somewhere; but I am cool with that. This is one thing, and may be Ecoris is not aware of the discussion there.
The way you have the statement for M, in this thread (Maths), does not imply that the probability distribution fades out as t increases a lot, which for example is implied (?) in the picture in the "Not in the Manual" thread. So, for completeness, let me put here the picture that you have:
What you say, is that M tends to be 0, rather than 1. This is interpreted as, there are a few second-ticks scattered in that time interval for which M(t) = 1, while for the vast majority of t's (whole seconds), it holds that M(t) = 0. So, there is no statement/guidance, that M(t) is concentrated more around specific intervals, rather than on others. Moreover, there is no information about the "dead zone" that you have in the picture. For instance, if the "dead zone" is more than 5 minutes, then, you can say that the probability for the first guy to see the bulb On is zero, because of the dead zone and the fact that the guy with probability 1 will enter into the room within the first 5 minutes. Which brings us to another assumption that is made for the solution of the problem. Forget about the distribution M(t) for a while. What is the distribution V(t) which determines when the guy will actually enter (visit) the room ? Of course here we can not say much (yet!), so the reasonable assumption is that V(t) is uniformly distributed in the same time interval as M(t). Again, this is because of the statement "a guy enters at random between 10 and 11". But let's forget V(t) again. Can you say something more about M(t) without giving away possibly all the details that you might want to keep for yourself; if any such details ?
I don't know if I am clear enough for everyone. We are interested in M and talking about M, but the picture that we have above is not really M (I emphasize this - although you wrote it too - because it may be confused). M can only be pulses of 0's and 1's if you want to go all the way down and exploit the random numbers that come from the generator. So, my guess is that what you have in the picture is some more general randomized scheme (looks like a bunch of exponentials, "saw"-voltage {I am not sure if such a terminology exists}) which should describe the probability distribution if we forget the "flaws" and the specific numbers that the current RNG is giving you. Essentially you want to describe the code, without looking at the "random" numbers that come by and determine the outcome (and hence the guy inside the room knows) and this way we can say something general about the RNG or the strategy the users should follow to maximize their ... "luck" and ... "morale"!
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The empty set
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AlexSpl
Responsible
Supreme Hero
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posted July 20, 2010 09:35 AM |
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Edited by AlexSpl at 09:37, 20 Jul 2010.
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I'll edit this post as soon as I conduct more tests and, then, try to describe the problem thoroughly. Thank for sharing your point of view on this problem. I'll answer later on as now I don't have tools needed to make tests and conclusions.
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dimis
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posted July 20, 2010 09:49 AM |
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Edited by dimis at 10:13, 20 Jul 2010.
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That's fine. Another thing:
In the plot you want to show Pr[M] which is not very clear what this thing is.
For me the most reasonable interpretation is that you have a collection of k M's {M_i} for i = 1 up to k, and then you go over each second-interval and check all the M_i's whether they are 0 or 1 in that second (they can not be anything else, right ?), and you average their sum by k. So, in other words, the plot shows the probability to get morale at a specific second as a function of all the M_i's. If this is correct, this is also missing from the description. Here, the role of the M_i's is some sort of equivalent of our so-called "skill-trees". But if this is true, then, there is still no description at all about the behavior of the M_i's (which should really be 1 M as you imply) regardless of the specific random numbers that determine the outcome.
Anyway, probably all this work should go into a separate thread. I think either Binabik or Ecoris have a related thread for Luck/Morale (I will check and let you know), or yours that hosts the "Oracle" are more suitable for all these. Then, in the end you can post something in the "Not in the Manual" thread with results.
additions:
I was talking about these two threads
- Base damage calculation, by Ecoris, and
- Feeling lucky?, by Binabik,
which are inactive for the last 3+ years (yeah, we are getting old ...)
From a quick skim to recall what was going on there, they are mostly about Damage calculation. So, in that sense, they are not exactly your "Luck/Morale" combo. Anyway, the decision is yours, you can revive them (go through them if you haven't done so; they are good), post in your thread with the "Oracle", post here, or simply start a new thread in the library.
Alright, I am going to get some sleep now. I hope these comments were helpful.
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The empty set
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ihor
Supreme Hero
Accidental Hero
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posted July 23, 2010 10:12 AM |
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Edited by ihor at 10:14, 23 Jul 2010.
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Hey dimis,
You promised to reveal the book in the end of our list of puzzles.
So, are we near the end?
Edit: I will post a new puzzle in the near future, unless nobody else did that.
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dimis
Responsible
Supreme Hero
Digitally signed by FoG
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posted July 23, 2010 05:45 PM |
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Edited by dimis at 18:07, 23 Jul 2010.
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Kvant and Quantum
You are right; I forgot to reveal my source. So, my primary source was
Quantum Quandaries (Google Book entry for a preview).
Essentially, the book is a collection of 100 brain-teasers, each of which has appeared in the past in the Kvant/Quantum magazines. Quantum was a translation of the Soviet Kvant magazine, but both in the US as well as in Greece we have been so smart in recent years that nobody translates the magazine any more.
I guess from 1993 and on, Kvant circulates bimonthly, and in each issue there was a page with 5 brain-teasers. Here is for example the July/August issue of Kvant from last year where the column on Brainteasers is on page 33. Here is a sample of the american version.
Homepages:
- Kvant Archive. It has all the articles available for free (in Russian) from 1970 up until last year (but it keeps on updating as time goes by).
- Quantum @ NSTA. It hasn't been updated since 2001.
- Quantum @ Katoptro. They still have copies of the series in their bookstore in Athens.
Other books that have appeared based on Quantum and are in English:
- Quantoons, which has a collection of amazing cartoons designed by the Croatian Tomas Bunk. Essentially all these cartoons decorate problems in Physics, and the book (Quantoons) is really about problems in Physics, ideal for people aged 15-18. Here is a sample of the way his cartoons look like; seriously, some of them would be ideal for posters.
- Kvant Selecta: Algebra and Analysis, I
- Kvant Selecta: Algebra and Analysis, II
- Kvant Selecta: Combinatorics I
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The empty set
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ihor
Supreme Hero
Accidental Hero
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posted July 23, 2010 06:09 PM |
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Oh I know this magazine.
I clearly remember I read some techniques of assembling Rubik's cube in Kvant. The issue was near 1985 year.
These brainteasers are actually home olympiad for pupils of 6-8 grade.
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dimis
Responsible
Supreme Hero
Digitally signed by FoG
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posted August 02, 2010 11:09 PM |
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BBC News: The Art of Mathematics
We have at least one guy (william) who is a fan of fractals.
Article and Video
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The empty set
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This thread is pages long: 
