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Let $a,b,c,d,e,f$ be positive integers such that $$\frac{a}{b} < \frac{c}{d} <\frac{e}{f}.$$

Suppose $af - be = -1$. Show that $d\ge b + f$.

Tried everything. I introduced constants $K_{1},K_{2}$ etc between $\frac{a}{b} , \frac{c}{d},\frac{e}{f}$ so that I have an equality, which can somehow be related to the equality given to us. I tried to use every special identity/inequality which I know, but to no avail. The problem always seems to be eliminating $c$ or $d$ from the inequality. I have been at it for 2 hours. This problem featured in a Regional Olympiad, using which after many stages the team for the IMO is selected.

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    @yurnero Sorry. It is $-1$. Edited the question. – 2017-02-20
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    @MartinR Oh right. I deleted my answer here and posted it there as I believe my hint is a useful direction. – 2017-02-20
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    Searching with [Approach0](https://approach0.xyz/search/?q=%24%5Cfrac%7Ba%7D%7Bb%7D%20%3C%20%5Cfrac%7Bc%7D%7Bd%7D%20%3C%5Cfrac%7Be%7D%7Bf%7D%24%2C%20%24d%20%5Cge%20b%2Bf%24&p=1) should become mandatory :) – 2017-02-20
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    @MartinR Yea It seems as if people have already asked this question many times – 2017-02-20
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    @MartinR IMHO the answers to the one I replied to are more valuable, so I'm not certain which one shall be the "root" one (note that it need not be the oldest one). – 2017-02-20

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