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This is an IMO problem from 1988, problem 6. The book does not provide a proof of this part and it is eluding me.

Let $$\cdots \lt s''' \lt s'' \lt s' \lt s$$ all be nonnegative integers (a finite sequence), and let $k$ be a nonnegative integer such that $$\begin{align*} 0 &\leq s''\\ s'' &=\frac{(s')^2 - k}{s}\\ s'''&=\frac{(s'')^2 - k}{s'}\\ &\vdots \end{align*}$$

and $s''^2+s'^2$=k(s''s'+1) ,$s+s''=-s'k,ss''=s'^2-k$ Prove that $k$ is a perfect square.

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    Please make your titles informative as to the problem itself; that makes them useful for other users of this site, and easier to search for. Telling us about your mental state is simply not very useful in the title.2011-10-31
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    @ArturoMagidin - i think i have already put the useful search information under the tags2011-10-31
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    Your title is **not** informative. Frankly, how you feel about the problem is of little interest to anyone except yourself, and the tags don't help elucidate what is *in* the body; [algebra] and [contest-math] are too general to really be useful in searching. So, please, *make your* **titles** *informative*!2011-10-31
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    @ArturoMagidin - Thanks, i don't know if it is appropriate to put the whole question on the title.2011-10-31
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    Put enough information to make the title informative; that's your goal.2011-10-31
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    If the $s$ have to be nonnegative integers, then the sequence has to be finite. The nonnegative integers do not contain infinite strictly decreasing sequences.2011-10-31
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    @ArturoMagidin - Thanks, edited.2011-10-31
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    And if finite sequences are allowed, then $1<3<6$ with $k=3$ appears to be a counterexample.2011-10-31
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    Why not just rewrite it as $s_{i+1}$s_is_{i+2}=s_i^2-k$? Much easier. @Henning: Indeed, it is simple to find counterexamples with length-3 sequences. But I see one issue: how do we have an infinite sequence of integers bounded between $0$ and a terminal $s\in\mathbb{N}$? – 2011-10-31
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    @Henning Makholm - sorry, i don't aware something is missing, now it is edited2011-10-31
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    Can you tell us which IMO? Sometimes, with a little more information, it is possible to find these things on the web.2011-10-31
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    @GerryMyerson - edited2011-10-31
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    According to http://www.artofproblemsolving.com/Wiki/index.php/1988_IMO_Problems/Problem_6 IMO 1988 6 is a problem about $ab+1$ dividing $a^2+b^2$. I can see where maybe one way to solve it would lead to your question. The solution at the link doesn't go that way. IMO 1988 6 is so nice it keeps coming up and you can probably find many solutions around the web.2011-10-31
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    For example, http://projectpen.files.wordpress.com/2008/03/pen06s.pdf2011-10-31
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    @GerryMyerson - But how would you apply infinite decent to conclude the statement?2011-10-31
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    @Victor: Please don't add things like "Unsolved Problem" in the title. For one thing, it implies this is an **open problem** (nobody has every solved it), which is not the case. For another, if your intention is just to point out that nobody has solved it *here*, then we **know that** by the fact that no answers have been posted.2011-11-02

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