Inequality of 6 Positive Integers

Asked Feb 20 '17 at 12:28

Active Feb 20 '17 at 12:38

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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.

edited Feb 20 '17 at 12:35

yo'

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asked Feb 20 '17 at 12:28

The East Wind

1

@yurnero Sorry. It is $-1$. Edited the question. – The East Wind Feb 20 '17 at 12:32
@MartinR Oh right. I deleted my answer here and posted it there as I believe my hint is a useful direction. – yo' Feb 20 '17 at 12:41
Searching with Approach0 should become mandatory :) – Martin R Feb 20 '17 at 12:44
@MartinR Yea It seems as if people have already asked this question many times – The East Wind Feb 20 '17 at 12:44
@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). – yo' Feb 20 '17 at 12:44

Inequality of 6 Positive Integers

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