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wu :: forums    riddles    easy (Moderators: SMQ, Eigenray, ThudnBlunder, Grimbal, towr, william wu, Icarus)    Triangles « Previous topic | Next topic » Pages: 1  Reply Notify of replies Send Topic Print    Author  Topic: Triangles  (Read 348 times) ThudnBlunder wu::riddles Moderator Uberpuzzler The dewdrop slides into the shining Sea     Gender: Posts: 4489 Triangles   « on: Jan 1st, 2005, 9:46am » Quote Modify Prove that if a triangle has sides of length a,b,c then sides of length 1/(a+b), 1/(b+c), 1/(a+c) can form another. IP Logged THE MEEK SHALL INHERIT THE EARTH.....................................................................er, if that's all right with the rest of you. towr wu::riddles Moderator Uberpuzzler Some people are average, some are just mean.     Gender: Posts: 13730 Re: Triangles   « Reply #1 on: Jan 2nd, 2005, 12:02pm » Quote Modify :: Would it be sufficient to note that the sides become more average? If you keep repeating the mapping the triangle approaches an equilateral one.   Of course you could also prove it based on a < b+c and b < a+c and c < a+b,   which is a sufficient and necessary condition for non-negative a,b,c forming a triangle.   The same thing has to hold for the new three numbers so 1/(a+b) < 1/(a+c) + 1/(b+c) 1/(a+b) < [(a+c) + (b+c)]/[(a+c)(b+c)]   (a+c)(b+c)< (a+b+2c) * (a+b)   ab+(a+b)c+c2 < a2+2ab+2ac+ b2+2bc c2 < (a+b)c + a2+ab+b2 which is easily seen to be true since   c < (a+b)   and thus   c2 < (a+b)c (+something>0)   and the same goes for the other permutation of a,b,c in the above relations :: IP Logged Wikipedia, Google, Mathworld, Integer sequence DB ThudnBlunder wu::riddles Moderator Uberpuzzler The dewdrop slides into the shining Sea     Gender: Posts: 4489 Re: Triangles   « Reply #2 on: Jan 3rd, 2005, 12:30pm » Quote Modify Quote: Would it be sufficient to note that the sides become more average?   If you keep repeating the mapping the triangle approaches an equilateral one. A typically incisive observation from towr.    However, no such mapping was defined. IP Logged THE MEEK SHALL INHERIT THE EARTH.....................................................................er, if that's all right with the rest of you. towr wu::riddles Moderator Uberpuzzler Some people are average, some are just mean.     Gender: Posts: 13730 Re: Triangles   « Reply #3 on: Jan 3rd, 2005, 1:12pm » Quote Modify It may not be defined, but it's intuitively there.. IP Logged Wikipedia, Google, Mathworld, Integer sequence DB 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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