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wu :: forums    riddles    easy (Moderators: ThudnBlunder, SMQ, Grimbal, william wu, Icarus, towr, Eigenray)    Square and Sum the Odd Tans « Previous topic | Next topic » Pages: 1  Reply Notify of replies Send Topic Print    Author  Topic: Square and Sum the Odd Tans  (Read 460 times) TenaliRaman Uberpuzzler I am no special. I am only passionately curious.     Gender: Posts: 1001 Square and Sum the Odd Tans   « on: Dec 3rd, 2003, 11:33am » Quote Modify Prove tan2(1o) + tan2(3o) + tan2(5o) + .... + tan2(89o) is an integer.   « Last Edit: Dec 3rd, 2003, 11:35am by TenaliRaman » IP Logged Self discovery comes when a man measures himself against an obstacle - Antoine de Saint Exupery Eigenray wu::riddles Moderator Uberpuzzler     Gender: Posts: 1948 Re: Square and Sum the Odd Tans   « Reply #1 on: Dec 3rd, 2003, 1:51pm » Quote Modify You could explicitly construct a polynomial whose roots are the 45 terms in the sum, but I don't know if I'd call that "easy," so maybe there's a simpler way. IP Logged TenaliRaman Uberpuzzler I am no special. I am only passionately curious.     Gender: Posts: 1001 Re: Square and Sum the Odd Tans   « Reply #2 on: Dec 3rd, 2003, 10:11pm » Quote Modify you have the right approach and there is an easier way to go about.     Hint: (quite vague)ask De Moivre for help IP Logged Self discovery comes when a man measures himself against an obstacle - Antoine de Saint Exupery Eigenray wu::riddles Moderator Uberpuzzler     Gender: Posts: 1948 Re: Square and Sum the Odd Tans   « Reply #3 on: Dec 4th, 2003, 11:55pm » Quote Modify If t = (2n+1)o, then 0 = cos(90t) = Re[ ei90t ] = Re[ (cos t + isin t)90 ]  = [sum]k=045 (90C2k)(-1)kcos90-2kt sin2kt. Dividing by cos90t shows that x = tan2t is a root of the polynomial f(x) = [sum]k=045 (90C2k)(-1)kxk = 0. Since f can have at most 45 roots, its roots are exactly {tan2(2n+1)o, n=0..44}, which therefore sum to 90C2 = 4005, an integer. IP Logged TenaliRaman Uberpuzzler I am no special. I am only passionately curious.     Gender: Posts: 1001 Re: Square and Sum the Odd Tans   « Reply #4 on: Dec 5th, 2003, 5:36am » Quote Modify Awesome!!!   IP Logged Self discovery comes when a man measures himself against an obstacle - Antoine de Saint Exupery Barukh Uberpuzzler     Gender: Posts: 2276 Re: Square and Sum the Odd Tans   « Reply #5 on: Dec 5th, 2003, 6:35am » Quote Modify I agree with TenaliRaman, it's absolutely brilliant!   Eigenray, could you please write a bit about how did you arrive at this solution?   TenaliRaman, do you think this problem belongs to easy section? IP Logged TenaliRaman Uberpuzzler I am no special. I am only passionately curious.     Gender: Posts: 1001 Re: Square and Sum the Odd Tans   « Reply #6 on: Dec 5th, 2003, 11:31am » Quote Modify I think it does belong to easy section, doesn't it?   i mean if i could solve it in one and a half days, it should be easy. IP Logged Self discovery comes when a man measures himself against an obstacle - Antoine de Saint Exupery 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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