1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Equilateral triangle

  1. Jan 13, 2008 #1
    1. The problem statement, all variables and given/known data
    Show that the curve x^3+3xy+y^3=1 contains only one set of three distinct points A,B, and C, which are vertices of an equilateral triangle.

    2. Relevant equations



    3. The attempt at a solution
    I randomly starting plotting points and found that all of them fell on the line y=1-x except (-1,-1). So, I just need to prove that these are the only points that satisfy that equation. It turns out that that equation factors into a very useful form. See B1 http://www.unl.edu/amc/a-activities/a7-problems/putnam/-pdf/2006s.pdf
    My question is how would you "discover" that really nice form if you were taking the test?
     
  2. jcsd
  3. Jan 14, 2008 #2

    Shooting Star

    User Avatar
    Homework Helper

    The first insticnt would be to try to factorise.

    x^3 + y^3 +3xy -1
    = (x+y)^3 -3x^2y -3xy^2 +3xy -1
    = [(x+y)^3 -1] -3xy(x+y-1)
    = [(x+y-1){(x+y)^2 +(x+y) +1}] -3xy(x+y-1)

    Now you can take (x+y-1) common and the rest would follow.
    -----------------------------------------------------------------
    p.s.

    If a simpler method occurs to me, I'll immdtly let you know. By inspection, some roots can be found to be (-1,-1), (-1,2) and (2,-1). It's not derivable at (-1,-1). If we plot the three points, we can sense some trouble at (-1,-1).
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?