P.S.: http://www.wolframalpha.com/input/?i=real+solutions+%28a^3+%2B+b^3+%2B+%2843+-+a+-+b%29^3+%3D+17299^2%29<http://www.wolframalpha.com/input/?i=real+solutions+%28a%5E3+%2B+b%5E3+%2B+%2843+-+a+-+b%29%5E3+%3D+17299%5E2%29>
<http://www.wolframalpha.com/input/?i=real+solutions+%28a%5E3+%2B+b%5E3+%2B+%2843+-+a+-+b%29%5E3+%3D+17299%5E2%29> http://www.wolframalpha.com/input/?i=integer+solutions+%28a^3+%2B+b^3+%2B+%2843+-+a+-+b%29^3+%3D+17299^2%29<http://www.wolframalpha.com/input/?i=integer+solutions+%28a%5E3+%2B+b%5E3+%2B+%2843+-+a+-+b%29%5E3+%3D+17299%5E2%29> http://www.wolframalpha.com/input/?i=rational+solutions+%28a^3+%2B+b^3+%2B+%2843+-+a+-+b%29^3+%3D+17299^2%29<http://www.wolframalpha.com/input/?i=rational+solutions+%28a%5E3+%2B+b%5E3+%2B+%2843+-+a+-+b%29%5E3+%3D+17299%5E2%29> On Sat, Jul 16, 2011 at 15:54, Diego Gavinowich <[email protected]> wrote: > There are infinite (real) solutions for: > > a^3 + b^3 + (43 - a - b)^3 = 17299^2 > > Basically you can replace *b* with any number and you have a quadratic > equation (because a^3 will go away)... > Any quadratic equation has 2 solutions (might give the same value, and both > are complex or real). > > -3ba^2 + 129a^2 - 3ab^2 + 258ab - 5547a + 129b^2 - 5547b - 299175894 = 0 > > If *b* was a constant then we have: > > a^2(-3b+129) + a (-3b^2+258b-5547) + (129b^2 - 5547b - 299175894) = 0 > > (-3b^2+258b-5547)^2 - 4 * (-3b+129) * (129b^2 - 5547b - 299175894) >= 0 is > what we need to have both real solutions (if I didn't make any silly > mistake) and that has infinite solutions (*b* <= 43 or *b* >= ~722.943). > > Best, > Diego > > > > On Sat, Jul 16, 2011 at 15:32, Alfonso J. Ramos <[email protected]> wrote: > >> Excuse me (I used spanish): not in the integers, but apparently there is >> one in the reals: >> >> http://www.wolframalpha.com/input/?i=+a+%2B+b+%2B+c+%3D+43+and+a^3+%2B+b^3+%2B+c^3+%3D+17299^2<http://www.wolframalpha.com/input/?i=+a+%2B+b+%2B+c+%3D+43+and+a%5E3+%2B+b%5E3+%2B+c%5E3+%3D+17299%5E2> >> >> 2011/7/16 Alfonso J. Ramos <[email protected]> >> >> No en los enteros, pero al parecer si en los reales: >>> >>> http://www.wolframalpha.com/input/?i=+a+%2B+b+%2B+c+%3D+43+and+a^3+%2B+b^3+%2B+c^3+%3D+17299^2<http://www.wolframalpha.com/input/?i=+a+%2B+b+%2B+c+%3D+43+and+a%5E3+%2B+b%5E3+%2B+c%5E3+%3D+17299%5E2> >>> >>> >>> 2011/7/16 Luke Pebody <[email protected]> >>> >>>> No answers exist for a + b + c = 43 and a^3 + b^3 + c^3 = 17299^2, >>>> even allowing them all to be negative: >>>> http://ideone.com/FTm8l >>>> >>>> On Sat, Jul 16, 2011 at 6:28 PM, Leopoldo Taravilse >>>> <[email protected]> wrote: >>>> > You can solve it with no programming. >>>> > a+b+c = 43 && a^3 + b^3 + c^3 = 17299 >>>> > As 17299 == 3 mod 4 then as a^3,b^3 and c^3 can be 0 or 1 mod 4 then >>>> a, b >>>> > and c are odd. >>>> > 27^3 > 17299 so a < b < c <= 25 >>>> > 1^3 == 1 mod 5 >>>> > 2^3 == 3 mod 5 >>>> > 3^3 == 2 mod 5 >>>> > 4^3 == 4 mod 5 >>>> > 5^3 == 5 mod 5 (or 0^3 == 0 mod 5) >>>> > So as 17299 == 43 + 1 mod 5 at least one of a, b and c must be 2 mod 5 >>>> so >>>> > there are two options, 7 or 17. >>>> > Case 1 (7): a+b = 36 && a^3 + b^3 = 16956 >>>> > Case 2 (17) a+b = 26 && a^3 + b^3 = 12386 >>>> > Let's see Case 1 first: a and b can be 1, 3, 5, 9, 11, 13, 15, 17, 19, >>>> 21, >>>> > 23 or 25, but if a = 1, 3, 5 or 9 then b > 25 so a and b can be 11, >>>> 13, 15, >>>> > 17, 19, 21, 23 or 25. We can easily discard 13 and 23 because of >>>> modulo 5 >>>> > reasons so we have 11, 15, 17, 19, 21 or 25 so we can try with this >>>> six >>>> > numbers. >>>> > Case 2: a and b can be 1, 3, 5, 9, 11, 13, 15, 17, 19, 21 or 23, but >>>> we can >>>> > discard 3, 13 and 23 again so we have 1, 5, 9, 11, 15, 17, 19 and 21 >>>> to try. >>>> > >>>> > On Sat, Jul 16, 2011 at 6:54 AM, micke <[email protected]> >>>> wrote: >>>> >> >>>> >> What are the three positive integers whose sum is 43. and the sum of >>>> >> the cubes of three integers is square of the number 17299. >>>> >> >>>> >> -- >>>> >> You received this message because you are subscribed to the Google >>>> Groups >>>> >> "google-codejam" group. >>>> >> To post to this group, send email to [email protected]. >>>> >> To unsubscribe from this group, send email to >>>> >> [email protected]. >>>> >> For more options, visit this group at >>>> >> http://groups.google.com/group/google-code?hl=en. >>>> >> >>>> > >>>> > -- >>>> > You received this message because you are subscribed to the Google >>>> Groups >>>> > "google-codejam" group. >>>> > To post to this group, send email to [email protected]. >>>> > To unsubscribe from this group, send email to >>>> > [email protected]. >>>> > For more options, visit this group at >>>> > http://groups.google.com/group/google-code?hl=en. >>>> > >>>> >>>> -- >>>> You received this message because you are subscribed to the Google >>>> Groups "google-codejam" group. >>>> To post to this group, send email to [email protected]. >>>> To unsubscribe from this group, send email to >>>> [email protected]. >>>> For more options, visit this group at >>>> http://groups.google.com/group/google-code?hl=en. >>>> >>>> >>> >> -- >> You received this message because you are subscribed to the Google Groups >> "google-codejam" group. >> To post to this group, send email to [email protected]. >> To unsubscribe from this group, send email to >> [email protected]. >> For more options, visit this group at >> http://groups.google.com/group/google-code?hl=en. >> > > -- You received this message because you are subscribed to the Google Groups "google-codejam" group. 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