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let $A$ is a finite set ,and the element are positive integers,and let $$B=\{\dfrac{a+b}{c+d}|a,b,c,d\in A\}$$ show that $$|B|\ge 2|A|^2-1$$ where $|X|$ is define finite set$X$ numbers

This is a 2014 china TST .and I see this reslut is similar $$\cos{(2x)}=2\cos^2{x}-1$$

and for this problem I have find some usefull paper:http://www.math.tau.ac.il/~nogaa/PDFS/null2.pdf

and http://www.cs.elte.hu/~karolyi/cd2.pdf

and http://www.webpages.uidaho.edu/newton/math376/Spring02/reudavid3.pdf

and http://cds.cern.ch/record/904813/files/cer-002575022.pdf

But I can't prove my problem .Thank you

math110
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2 Answers2

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I think it is a quite recent result, I saw the Balog paper http://arxiv.org/abs/1402.5775 just a few days ago.

Jack D'Aurizio
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    Despite the fact that Balog's argument only relies on lines-counting and the $\frac{a}{b}\leq\frac{a+c}{b+d}\leq\frac{c}{d}$ inequality, I wonder about the Chinese attitude to put research-like problems in a TST... – Jack D'Aurizio Mar 17 '14 at 13:20
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Set $A+A=\{a+b|a,b\in A\}$. By the Cauchy-Davenport theorem, $$|A+A|\ge 2|A|-1$$ (equality is achieved if $A=\{1,2,\ldots,n\}$). Hence, we may consider $$B=\left\{\frac{a}{b}:a,b\in A+A\right\}$$

This gives $(2|A|-1)^2=4|A|^2-4|A|+1$ values for $B$, not all of which are distinct. It remains to count distinct elements of $B$.

vadim123
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