superprismatic
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The problem intent is that there are 4 differently sized concentric circles in the plane. I hope you enjoy working on this problem. Welcome to the Den! Your out-of-the-box way of looking at things will help you with many of the puzzles here.
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What conditions must be put on 4 concentric circles so that it is possible to draw a rectangle having one corner on each circle?
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From the form of your answer, it looks like you measured distance between points as being the length of arc of a great circle between two points. Is that correct? I just use the usual euclidean distance in 3-space. Either one is fine -- I just wanted to know which you used. I tried to get to your answer but I couldn't see how to get there. I would need a bit more of an explanation, including which distance measure you used. If you are correct, your answer would beat my answer by a bit (how much depends on the metric you used).
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Am I missing something here? I'm sorry, I goofed. You could do this with any circle on the surface of the sphere. A zig-zag pattern encircling the sphere would work as well. Alas! I neglected to add a crucial word to the OP. I corrected it in post #9.
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Woops! I left out an important word in the OP. I should have asked for the maximum value of D. Here's the corrected problem: If we spread out 24 points on the surface of a unit sphere (i.e., a sphere with radius 1) so that each point is a distance D away from its nearest neighbour, what would be the maximum value of D?
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If we spread out 24 points on the surface of a unit sphere (i.e., a sphere with radius 1) so that each point is a distance D away from its nearest neighbour, what would be the value of D?
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Nice Job! I'm glad you didn't write me to verify because it was quite difficult to make up this puzzle without mistakes. It was even harder to insure that there was only one solution!
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In the following puzzle, each of the capital letters A, B, C, D, E, F, G, H, and I stands for a unique digit from 1 to 9. Concatenated capital letters stand for the appropriate contatenated digits, so, for example, if A=7 and B=3, then AB would equal 73. The powers of x in the following polynomials are not necessarily written in decreasing order. If the polynomial CxG - AAxD + ABxC + AExB - A is divided by AxA + FxG + BxB - D, the quotient would be ExH - IxA + G and the remainder would be BDxE + AFxH - HHxA + AA. Find the values of A, B, C, D, E, F, G, H, and I.
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Sorry, it would take me far too long to write out a cogent explanation. I used a Linear Algebra approach although there are many others (Fourier Analysis being one of them). I can't see any way to explain it simply.
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I'm so sorry, I must have answered too hastily. Yes, Smith surely got it right as well. Thanks for pointing it out to me.
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I think that, in your solution, there are 4 confused pigs: E2, D3, C4, and F5.
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philip1882, k-man, and rlgandy got it right. rlgandy had the simplest explanation.
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You have it. Good job! Your first two sentences had the critical observation.
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You got it! I was just about to say that your last post was close (but no cigar) when I saw you posted this. Nice recurrence method, too!
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What is the coefficient of x45 in the power series expansion of (1-x)-1(1-x3)-1(1-x9)-1(1-x15)-1?
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Let N be any positive integer. Is it true that there is always at least one number in the set {N,N+1,N+2,N+3,N+4} which is relatively prime to the product of the other four?
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A regular polygon has 1023 sides, each of length 1. Is the area between the inscribed and circumscribed circles of this polygon greater than 1?
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araver's got it! If anyone is still interested in this problem but doesn't want to follow in araver's footsteps, here is a slightly different but related problem: Find the smallest degree trinomial multiple of P[x] all of whose nonzero coefficients are 1.
