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wu :: forums    riddles    easy (Moderators: towr, ThudnBlunder, Eigenray, william wu, SMQ, Icarus, Grimbal)    X marks the spot « Previous topic | Next topic » Pages: 1  Reply Notify of replies Send Topic Print    Author  Topic: X marks the spot  (Read 2947 times) towr wu::riddles Moderator Uberpuzzler Some people are average, some are just mean.     Gender: Posts: 13730 X marks the spot   « on: Nov 4th, 2012, 10:15am » Quote Modify (Also from "professor Stewart's hoard of mathematical treasures", but much abbreviated)     A, B and C are three consecutive points on a 140 x 140 square, inside the square lies point X, the distance BX is 99. Find the integer distances AX and CX (with AX < CX)     A given hint: if the sum of two squares is 0 modulo 7, then so are both squares.     Bonus points if you can find a square with integer sides and an internal point with integer distances form all four corners. (NB; probably hard) « Last Edit: Nov 4th, 2012, 10:16am by towr » IP Logged Wikipedia, Google, Mathworld, Integer sequence DB pex Uberpuzzler     Gender: Posts: 880 Re: X marks the spot   « Reply #1 on: Nov 4th, 2012, 12:03pm » Quote Modify on Nov 4th, 2012, 10:15am, towr wrote: Bonus points if you can find a square with integer sides and an internal point with integer distances form all four corners. (NB; probably hard) Probably, yes... It is not known if there exists a point in a unit square all of whose distances from the corners are rational. « Last Edit: Nov 4th, 2012, 12:03pm by pex » IP Logged towr wu::riddles Moderator Uberpuzzler Some people are average, some are just mean.     Gender: Posts: 13730 Re: X marks the spot   « Reply #2 on: Nov 4th, 2012, 12:36pm » Quote Modify Well, it's worth a try. IP Logged Wikipedia, Google, Mathworld, Integer sequence DB pex Uberpuzzler     Gender: Posts: 880 Re: X marks the spot   « Reply #3 on: Nov 4th, 2012, 12:44pm » Quote Modify Hasn't stopped me from trying, of course IP Logged jollytall Senior Riddler     Gender: Posts: 585 Re: X marks the spot   « Reply #4 on: Nov 5th, 2012, 10:39am » Quote Modify No elegant solution, but 85,113 seems right. IP Logged towr wu::riddles Moderator Uberpuzzler Some people are average, some are just mean.     Gender: Posts: 13730 Re: X marks the spot   « Reply #5 on: Nov 5th, 2012, 11:11am » Quote Modify Yup, those are correct.   I solved it using the law of cosines as intermediate step, and professor Stewart used an intermediate step of right-angled triangles.   In both cases that gives (29400 - a2)2 + (29400 - c2)2 = 277202. Using the hint, you can narrow it down to a few options. I worked modulo 77, which narrows it down further. Because the trick works for both 7 and 11 and therefore 77. Though I did mess up with taking square roots modulo 77. « Last Edit: Nov 5th, 2012, 11:12am by towr » IP Logged Wikipedia, Google, Mathworld, Integer sequence DB marlonmark Newbie     Gender: Posts: 43 Re: X marks the spot   « Reply #6 on: Jan 6th, 2013, 10:29pm » Quote Modify I think its 85113 times IP Logged towr wu::riddles Moderator Uberpuzzler Some people are average, some are just mean.     Gender: Posts: 13730 Re: X marks the spot   « Reply #7 on: Jan 7th, 2013, 9:10am » Quote Modify That's not an answer in any way related to this question. Are you just spamming? IP Logged Wikipedia, Google, Mathworld, Integer sequence DB Annettagiles Junior Member     Gender: Posts: 67 Re: X marks the spot   « Reply #8 on: Oct 7th, 2014, 4:21am » Quote Modify 46x may be the right answer I think so. 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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