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Let $a$, $b$, $c$, $d$ and $d$ be non-negative numbers such that $a^2+b^2+c^2+d^2+e^2=5$. Prove that: $$\frac{1}{7-2a}+\frac{1}{7-2b}+\frac{1}{7-2c}+\frac{1}{7-2d}+\frac{1}{7-2e}\leq1$$

The equality occurs also for $a=2$ and $b=c=d=e=\frac{1}{2}$.

I tried TL: $$1-\sum_{cyc}\frac{1}{7-2a}=\sum_{cyc}\left(\frac{1}{5}-\frac{1}{7-2a}\right)=\frac{2}{5}\sum_{cyc}\frac{1-a}{7-2a}=$$ $$=\frac{2}{5}\sum_{cyc}\left(\frac{1-a}{7-2a}+\frac{a^2-1}{10}\right)=\frac{1}{25}\sum_{cyc}\frac{(a-1)^2(3-2a)}{7-2a}$$ and I don't see what is the rest.

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    Where did you find the problem? – 2017-02-23

1 Answers 1

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We must to prove that $f(a_1)+f(a_2)+f(a_3)+f(a_4)+f(a_5) \le 1$ ,

for non-negative $a_1+a_2+a_3+a_4+a_5=5$

$$f(x)=\dfrac{1}{7-2\sqrt{x}}$$

Since $f''(x)=\dfrac{(7-6\sqrt{x})}{2(2x-7\sqrt{x})^3}$ , we only need to consider the inequality in case $0< a_1=a_2=a_3=a_4=t^2 \le 1 \ , \ a_5=5-4t^2$

Clearly

$g(t)=\dfrac{4}{7-2t}+\dfrac{1}{7-2\sqrt{5-4t^2}} $

$ g'(t)=\dfrac{8}{(7-2t)^2}-\dfrac{8t}{\sqrt{5-4t^2}(7-2\sqrt{5-4t^2})^2}\ge 0 \Leftrightarrow $

$\sqrt{5-4t^2}\cdot(69-16t^2) \ge 4t^3-140t^2+49t+140>0 \Leftrightarrow$

$(t-1)(2t-1)(t+1)(104t^3-60t^2+1062t-841) \le 0 \ ,\ t\in [0,1]$

So maximum of $g(t)$ is attained at $t_1=\dfrac{1}{2}$ and $t_2=1$.

$$g(t_1)=g(t_2)=1$$

Equality holdes for : $(a=b=c=d=e=1) $ and $\left( a=b=c=d=\dfrac{1}{2}, e=2\right)$

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    Yes, you are right! I also proved this problem today with the same start, but I just was very busy. We can use also the Vasc's LCF Theorem. Thank you! – 2017-02-23
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    Please pardon my ignorance, but why is it sufficient to consider the case $0< a_1=a_2=a_3=a_4=t^2 \le 1 \ , \ a_5=5-4t^2$ ? – 2017-02-23
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    @Martin R Since $f''(x)<0$ for all $0$x_1=x_2=x_3=x_4$. It's the Vasc's LCF Theorem. – 2017-02-24
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    @MichaelRozenberg: Thanks! – I did not know that theorem before, and found it only somewhat hidden on AOPS: https://www.artofproblemsolving.com/community/c6h32332p202112. Is there a better reference with proof? What do you think about adding an entry to http://math.stackexchange.com/tags/inequality/info? – 2017-02-24
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    @Martin R 5 The Vasc's proof of this Theorem is very nice! It's a very interesting play with Jensen and Karamata. V.Cirtoaje "Algebraic inequalities" p. !43-149. Now about adding. I fixed this taginfo year ago, , but someone deleted my work. – 2017-02-25