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the sum of two positive numbers equals the difference of the squares of the two numbers, which equals the quotient of the larger number when divided by the smaller. What is the smaller number? Provide the exact answer.

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closed as off-topic by Gareth McCaughan, Rand al'Thor, Will, Deusovi Oct 2 '16 at 23:13

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  • "This question is off-topic as it appears to be a mathematics problem, as opposed to a mathematical puzzle. For more info, see "Are math-textbook-style problems on topic?" on meta." – Gareth McCaughan, Rand al'Thor, Will, Deusovi
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    $\begingroup$ It looks like sqrt{2}/2. $\endgroup$ – Matsmath Oct 2 '16 at 21:09
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This seems more textbook problem than puzzle, but:

Let our numbers be $a,b$ where $a<b$. Then we have $a+b=b^2-a^2$, whence either $a+b=0$ (nope, they're positive numbers) or $1=b-a$. And now $a+b=b/a$ means, substituting $b=a+1$, that $2a+1=1+1/a$ or $2a^2=1$ so (since our numbers are positive) $a=1/\sqrt{2}$.

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  • $\begingroup$ I was wondering if there is a geometric interpretation of this puzzle. Although it must be quite difficult to find an insightful one. $\endgroup$ – Matsmath Oct 2 '16 at 21:43
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    $\begingroup$ To get a difference of two squares out of a geometrical construction you need to do something fairly nontrivial, so I'm not very hopeful. Though there are some nice geometrical constructions to, e.g., show that sqrt(2) is irrational by an infinite descent, which I guess is the sort of thing you're thinking of. $\endgroup$ – Gareth McCaughan Oct 2 '16 at 21:51

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