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wu :: forums    riddles    medium (Moderators: Grimbal, towr, william wu, ThudnBlunder, Icarus, SMQ, Eigenray)    Consider The Function, Get The Values « Previous topic | Next topic » Pages: 1  Reply Notify of replies Send Topic Print    Author  Topic: Consider The Function, Get The Values  (Read 1519 times) K Sengupta Senior Riddler     Gender: Posts: 371 Consider The Function, Get The Values   « on: Jun 27th, 2007, 7:43am » Quote Modify g is a function possessing the  domain and range of the positive integers satisfying:     (1) g(p+1) > g(p);     (2) g(g(p)) = 3p   Analytically determine all possible values that g(955) can take. IP Logged pex Uberpuzzler     Gender: Posts: 880 Re: Consider The Function, Get The Values   « Reply #1 on: Jun 27th, 2007, 7:50am » Quote Modify A quick clarification: is "range" supposed to be "codomain"? Otherwise, no such function exists: an increasing function with domain and range the positive integers has  to have g(p) = p. Then g(1) = 1 and by your second rule, g(1) = 3 - contradiction. IP Logged pex Uberpuzzler     Gender: Posts: 880 Re: Consider The Function, Get The Values   « Reply #2 on: Jun 27th, 2007, 8:09am » Quote Modify Assuming that this is the case:   hidden: I ruled out g(1) = 1 above. Let g(1) = k > 1. Then g(k) = 3. As 1 < k, by the monotonicity condition, we must have g(1) < g(k) => k < 3. Thus, k = 2.   This forces the following values: g(1) = 2; g(2) = 3; g(3) = 6; g(6) = 9; g(9) = 18; g(18) = 27; g(27) = 54; g(54) = 81; g(81) = 162; g(162) = 243; g(243) = 486; g(486) = 729; g(729) = 1458; g(1458) = 2187.   Notice that 1458 - 729 = 2187 - 1458. Again, by monotonicity, we must have g(p) = p + 729 for all 729 < p < 1458. In particular, g(955) = 1684.   Nice problem!   Edit: stupid smilies... « Last Edit: Jun 27th, 2007, 8:09am by pex » IP Logged peoplepower Junior Member     Posts: 63 Re: Consider The Function, Get The Values   « Reply #3 on: Jul 29th, 2012, 10:06am » Quote Modify Note that if 0 <= k<2*3m, then g(3m+k)=(2*3m+k)[k<3m]+(3m+1+3k)[k>=3m ] (using Iverson brackets), following a proof by induction on m. « Last Edit: Jul 29th, 2012, 10:07am by peoplepower » 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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