Refer to the earlier problem.

Can you find all possible non zero real quadruplet(s) (P, Q, R, S) that satisfy the following system of simultaneous equations?

2*Q = P + 19/P, and:

2*R = Q + 19/Q, and:

2*S = R + 19/R, and:

2*P = S + 19/S.

Bonus Question:

In the problem given above, if the number 19 was replaced throughout by a real constant M > 1, what would have been the possible non zero real quadruplet(s) (P, Q, R, S) that satisfy the given set of equations, in terms of M?

a) Well, one possibility is P = Q = R = S.
In that case, 2*P = P + M/P, and P = sqrt(M) or -sqrt(M)

b) Another possibility is P = R, Q = S, not necessarily equal.
But then,
  2Q = P + M/P
  2P = Q + M/Q

  2PQ = P*P + M
  2PQ = Q*Q + M,

  so P = Q = R = S, and this is our first case

c) Are there solutions where P, Q, R, and S are all different?
I haven't worked that part out, but I wouldn't be surprised either way.

Posted by Steve Herman on 2008-05-28 13:56:14