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        riddles >> medium >> Absolute And Constant Puzzle
        
(Message started by: K Sengupta on Jun 12^th, 2008, 7:33am)

Title: Absolute And Constant Puzzle
Post by K Sengupta on Jun 12^th, 2008, 7:33am

The pair (P, Q) of real numbers satisfy the following system of equations:

2^|P| + |P| = Q + P² + M, and P² + Q² = 1 ………(*)
where M is a real constant.

Determine all possible values of M such that the system of equations in (*) admits of precisely one solution in (P, Q).

Note: |P| denotes the absolute value of P.

Title: Re: Absolute And Constant Puzzle
Post by pex on Jun 12^th, 2008, 8:33am

on 06/12/08 at 07:33:43, K Sengupta wrote:

I think only [hide]M = 0, (P, Q) = (0, 1)[/hide] and [hide]M = 2, (P, Q) = (0, -1)[/hide] work, because [hide]P only appears through |P| and P², making P and -P interchangeable[/hide].

Title: Re: Absolute And Constant Puzzle
Post by Eigenray on Jun 12^th, 2008, 9:38am

m=[hide]0,2[/hide] are the only values for which [hide]there are an odd number of solutions[/hide], but only m=[hide]0[/hide] has a unique solution.

Title: Re: Absolute And Constant Puzzle
Post by SMQ on Jun 12^th, 2008, 9:43am

Only [hide]M = 0[/hide] has a unique solution in both P and Q: [hide]P = 0, Q = 1[/hide]. As pex points out, [hide]P must be zero (and therefore Q = +/-1) or there are multiple solutions in P[/hide], but also [hide]M must be < 2 or there are multiple solutions in Q, for instance M = 2 can be solved by P = +/-1, Q = 0 as well as P = 0, Q = -1.[/hide]

([hide]The maximum value of M (around 2.557) clearly does have a unique solution in Q but it admits two solutions in P with opposite signs.[/hide])

Edit: dang, ninja'd by Eigenray. ;)

--SMQ

Title: Re: Absolute And Constant Puzzle
Post by pex on Jun 12^th, 2008, 9:56am

:-[

Okay, maybe I should think a bit more before posting.