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riddles >> hard >> An Astute Integer Triplet Puzzle
(Message started by: K Sengupta on Mar 11th, 2007, 5:19am)

Title: An Astute Integer Triplet Puzzle
Post by K Sengupta on Mar 11th, 2007, 5:19am
Analytically determine all possible integer triplets  (x ,y, z) satisfying this equation:

sqrt(x+ sqrt(x+ sqrt(x + sqrt(x ......+ sqrt(x + sqrt(x + y))))) = z;
with |x| < 100,000 and |y|< 100,000; where:
|n|
= n for n>=0 and;
= -n for n<0.

The square root symbol in the above relationship is repeated 2007 times.

Title: Re: An Astute Integer Triplet Puzzle
Post by Barukh on Mar 14th, 2007, 12:24am
It depends on the interpretation of the sqrt...

In its positive sense, my guess is that [hide](a2 - a, a, a)[/hide] is the only family of solutions.

Title: Re: An Astute Integer Triplet Puzzle
Post by Aryabhatta on Mar 14th, 2007, 12:01pm
I guess we do an induction on the number of times the square root repeats and show that the solutions described by Barukh are the only ones...

For an n square root equation, It would be easy to prove that sqrt(x+y) is also an integer and then replace sqrt(x+y) with u
and solve the same equation with with n-1 square roots with y replaced by u.

btw, why do you insist on using adjectives like "Astute", "Ingenious" etc ?

Title: Re: An Astute Integer Triplet Puzzle
Post by Barukh on Mar 14th, 2007, 11:00pm

on 03/14/07 at 12:01:35, Aryabhatta wrote:
btw, why do you insist on using adjectives like "Astute", "Ingenious" etc ?

Maybe, because this makes the problems more Aesthetic?  ;)

Title: Re: An Astute Integer Triplet Puzzle
Post by ThudanBlunder on Mar 15th, 2007, 12:31am

on 03/14/07 at 12:01:35, Aryabhatta wrote:
btw, why do you insist on using adjectives like "Astute", "Ingenious" etc ?

Besides the fact that the number of letters in each of the words 'magic', 'astute', 'unusual', 'greatest', 'ingenious', and 'analytical' are in arithmetic progression, perhaps because an appropriate juxtaposition of judiciously chosen adjectives confers less ambiguity upon the expressions constituting the thread titles, thus reducing the probability of anomalous interpretations of the problems under reference.

 :D

Title: Re: An Astute Integer Triplet Puzzle
Post by Aryabhatta on Mar 15th, 2007, 12:52pm

on 03/15/07 at 00:31:56, ThudanBlunder wrote:
Besides the fact that the number of letters in the words 'astute', 'integer', 'greatest', 'ingenious', and 'trivariate' are in arithmetic progression, perhaps because a suitable juxtaposition of appropriate adjectives bestows less ambiguity on the expressions constituting the thread titles, thus reducing the probability of anomalous interpretations of the problems under reference.

:D


Lol!
I guess that must be it.  :P


Please take no offense KSengupta. There is no such intention.

Title: Re: An Astute Integer Triplet Puzzle
Post by ThudanBlunder on Mar 15th, 2007, 6:09pm

on 03/15/07 at 12:52:05, Aryabhatta wrote:
Please take no offense KSengupta. There is no such intention.

Of course not, KS.  

Title: Re: An Astute Integer Triplet Puzzle
Post by K Sengupta on Mar 22nd, 2007, 8:07am

on 03/15/07 at 00:31:56, ThudanBlunder wrote:
Besides the fact that the number of letters in each of the words 'magic', 'astute', 'unusual', 'greatest', 'ingenious', and 'analytical' are in arithmetic progression, perhaps because an appropriate juxtaposition of suitable adjectives confers less ambiguity upon the expressions constituting the thread titles, thus reducing the probability of anomalous interpretations of the problems under reference.

:D


Thanks for the vote of confidence TB.

I may add that,  the said methodology also facilitates the proceedure corresponding to reduction of  the probability of recrudescence of a multiplicity of threads with the same title,  but possessing different concepts, which I understand is at direct variance with the provisions governing this website.



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