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wu :: forums    riddles    putnam exam (pure math) (Moderators: Icarus, SMQ, towr, Eigenray, william wu, Grimbal)    Radicals « Previous topic | Next topic » Pages: 1  Reply Notify of replies Send Topic Print    Author  Topic: Radicals  (Read 840 times) ThudnBlunder Uberpuzzler The dewdrop slides into the shining Sea     Gender: Posts: 4489 Radicals   « on: Apr 2nd, 2007, 1:49am » Quote Modify Find all positive numbers a such that 3(3 + a) + 3(3 - a) is an integer.   « Last Edit: Apr 2nd, 2007, 1:52am by ThudnBlunder » IP Logged THE MEEK SHALL INHERIT THE EARTH.....................................................................er, if that's all right with the rest of you. Barukh Uberpuzzler     Gender: Posts: 2276 Re: Radicals   « Reply #1 on: Apr 2nd, 2007, 3:14am » Quote Modify Should a be an integer? IP Logged Grimbal wu::riddles Moderator Uberpuzzler     Gender: Posts: 7527 Re: Radicals   « Reply #2 on: Apr 2nd, 2007, 4:59am » Quote Modify That would leave only the numbers 0 to 9 to try.  I would say any number.   "Positive" suggests a can not be complex.  So any positive real number.   There might be extra solutions if the result can be any Gaussian integer. IP Logged towr wu::riddles Moderator Uberpuzzler Some people are average, some are just mean.     Gender: Posts: 13730 Re: Radicals   « Reply #3 on: Apr 2nd, 2007, 7:41am » Quote Modify Isn't the answer all positive numbers? Like last time?   For some integer x, and some y we have [x+ sqrt(y)]3=3+sqrt(a) and [x -  sqrt(y)]3=3-sqrt(a)   x+ sqrt(y) + x -  sqrt(y) = 2 x « Last Edit: Apr 2nd, 2007, 7:43am by towr » IP Logged Wikipedia, Google, Mathworld, Integer sequence DB Grimbal wu::riddles Moderator Uberpuzzler     Gender: Posts: 7527 Re: Radicals   « Reply #4 on: Apr 2nd, 2007, 10:03am » Quote Modify I don't think you can always find such x and y.   Consider a=0. 3(3+a) + 3(3-a) = 2·33 and it is not an integer. IP Logged SMQ wu::riddles Moderator Uberpuzzler     Gender: Posts: 2084 Re: Radicals   « Reply #5 on: Apr 2nd, 2007, 11:57am » Quote Modify In fact, since a = 0 gives a value of 233 2.88, and the function is monotonic decreasing and always positive, the only integer values it can have are 2 and 1 at the corresponding values of a, right?   --SMQ IP Logged --SMQ Barukh Uberpuzzler     Gender: Posts: 2276 Re: Radicals   « Reply #6 on: Apr 2nd, 2007, 12:06pm » Quote Modify SMQ, I think you are right.   The problem is to find an expression (I think in radicals) for such a. IP Logged SMQ wu::riddles Moderator Uberpuzzler     Gender: Posts: 2084 Re: Radicals   « Reply #7 on: Apr 2nd, 2007, 12:24pm » Quote Modify For x = 3(3 + a) + 3(3 - a), I get a = 9 - (x3 - 6)3/(3x)3, which gives a = 242/27 at x = 2 and a = 368/27 at x = 1.   Edit -- derivation: (1) x = 3(3 + a) + 3(3 - a)   (2) x3 = 6 + 33((3 + a)2(3 - a) + 33((3 + a)(3 - a)2)   (3) (x3 - 6)/3 = 3((3 + a)2(3 - a) + 3((3 + a)(3 - a)2)   (4) ((x3 - 6)/3)3 = (3 + a)2(3 - a) + 33((3 + a)5(3 - a)4) + 33((3 + a)4(3 - a)5) + (3 + a)(3 - a)2   (5) ((x3 - 6)/3)3 = (3 + a)(3 - a)[6 + 33((3 + a)2(3 - a) + 33((3 + a)(3 - a)2)]   (6) ((x3 - 6)/3)3 = (9 - a)[x3] (substituting from (2) for the bracketed term)   (7) 9 - a = ((x3 - 6)/(3x))3     --SMQ « Last Edit: Apr 2nd, 2007, 1:29pm by SMQ » IP Logged --SMQ Eigenray wu::riddles Moderator Uberpuzzler     Gender: Posts: 1948 Re: Radicals   « Reply #8 on: Apr 2nd, 2007, 4:31pm » Quote Modify There's a simpler (or at least easier to read) derivation:   If x = u + v, then x3 = u3 + v3 + 3uv x. Substituting then, we have x3 = 6 + 3(9-a)1/3 x, which is easily solved for a. IP Logged Barukh Uberpuzzler     Gender: Posts: 2276 Re: Radicals   « Reply #9 on: Apr 4th, 2007, 5:52am » Quote Modify   How about this variation: 3(3 + a) - 3(3 - a) is an integer, and a is a rational? IP Logged SMQ wu::riddles Moderator Uberpuzzler     Gender: Posts: 2084 Re: Radicals   « Reply #10 on: Apr 4th, 2007, 6:39am » Quote Modify By Eigenray's derivation, x = u - v x3 = u3 - v3 - 3uvx a = 9 - (6 - x3)/(3x)[/sup]3[/sup] so a is rational (and positive) for all integer x 0.   Edit: erm, not quite, as Barukh points out below.  See, this is why I'm a computer programmer and not a mathematician ... other programmers find my off-the-cuff answers convincing.     --SMQ « Last Edit: Apr 4th, 2007, 10:59am by SMQ » IP Logged --SMQ Barukh Uberpuzzler     Gender: Posts: 2276 Re: Radicals   « Reply #11 on: Apr 4th, 2007, 10:49am » Quote Modify on Apr 4th, 2007, 6:39am, SMQ wrote: x = u - v x3 = u3 - v3 - 3uvx[/hide]. What's the value of u3 - v3?   IP Logged Pages: 1  Reply Notify of replies Send Topic Print Forum Jump: ----------------------------- riddles -----------------------------  - easy   - medium   - hard   - what am i   - what happened   - microsoft   - cs => putnam exam (pure math)   - suggestions, help, and FAQ   - general problem-solving / chatting / whatever ----------------------------- general -----------------------------  - guestbook   - truth   - complex analysis   - wanted   - psychology   - chinese « Previous topic | Next topic »

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