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wu :: forums    riddles    medium (Moderators: towr, ThudnBlunder, Icarus, william wu, Eigenray, Grimbal, SMQ)    solve 4 « Previous topic | Next topic » Pages: 1  Reply Notify of replies Send Topic Print    Author  Topic: solve 4  (Read 1140 times) tony123 Junior Member     Posts: 61 solve 4   « on: Jul 4th, 2007, 6:27am » Quote Modify http://www.sosmath.com/CBB/latexrender/pictures/a265198aee04e6f1c530d1d3 93b4985e.gif IP Logged Sir Col Uberpuzzler impudens simia et macrologus profundus fabulae     Gender: Posts: 1825 Re: solve 4   « Reply #1 on: Jul 4th, 2007, 7:53am » Quote Modify :: Let S = (61/2-51/2)x + (61/2+51/2)x + (31/2-21/2)x + (31/2+21/2)x   (a-b)-1 = 1/(a-b) = (a+b)/((a-b)(a+b)) = (a+b)/(a2-b2)   If a2 = b2+1, then (a-b)-1 = a+b.   Therefore (61/2-51/2)-x = (61/2+51/2)x and (61/2+51/2)-x = (61/2-51/2)x.   Hence S be the same value for x = -x.   Considering positive x, it is clear that as x decreases S decreases and as x increases S also increases, thus for S = n there is only one solution in positive x.   (u-v)2 + (u+v)2 = 2(u2+v2)   Let a = 61/2, b=51/2, c=31/2, and d=21/2.   So (a-b)2 + (a+b)2 + (c-d)2 + (c+d)2 = 2(6+5) + 2(3+2) = 32.   Hence x = -2,2. :: IP Logged mathschallenge.net / projecteuler.net 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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