En, p 415........................................................................................................................
Fr, p 418.....................................................................................................................
ANSWER CiP
Equality \Leftrightarrow x_{1}=\cdots=x_{n}=16
SOLUTION CiP
LEMMA If 0\leqslant x <1 then
(1) \frac{1}{1-x}\geqslant 1+4x^{2}.
Equality if and only if x=0 or x=\frac{1}{2}.
Proof of Lemma
We perform simple calculation
\frac{1}{1-x}-1-4x^{2}=\frac{x(1-2x)^{2}}{1-x} \geqslant 0.
The conclusions result.
\square (LEMMA)
Let f(x)=\frac{1}{8-\sqrt{x}} where 0<x<64. If we write
f(x)=\frac{1}{8} \cdot \frac{1}{1-\frac{\sqrt{x}}{8}} \geqslant \frac{1}{8} \cdot(1+4(\frac{\sqrt{x}}{8})^{2})
where I applied the Lemma for \frac {\sqrt{x}}{8} instead of x, then it turns out
(2) f(x) \geqslant \frac{1}{8}+\frac{x}{128}.
We put in the formula (2), one by one x=x_{1}, x_{2}, \cdots x_{n} and add the obtained results.
So we have
\sum_{i=1}^{n}f(x_{i})\geqslant \sum_{i=1}^{n}\left ( \frac{1}{8}+\frac{x_{i}}{128} \right )=\frac{n}{8}+ \frac{\sum_{i=1}^{n}x_{i}}{128}=\frac{n}{8}+\frac{16 \cdot n}{128}=\frac{n}{8}+\frac{n}{8}=\frac{n}{4}
what needed to be shown.
\blacksquare
The equal sign occur when in (2) eqal signs occur, i.e. \frac{\sqrt{x_{i}}}{8}=\frac{1}{2}, so all x_{i}=16.
Raspunsul de confirmare
in Crux Mathematicorum. Our file server has received your submission
and the appropriate editor will be reviewing it in sequence or as
needed.
Below is a receipt confirming the submission you sent us.
Tracking Number: 10788
Received: Mon Oct 12 16:50:40 2020
From: Ciobanu Petre
Scoala Gimnaziala "Samuil Micu" SADU
Sibiu, Romania
Email: ptr.ciobanu@gmail.com
Type: Solve a Crux (Numbered) Problem
(problem 4574)
Files:
yynwlsnl.pdf
Comments:
See my Blog
https://ogeometrie-cip.blogspo
| lun., 12 oct., 19:52 | ||
|
=============
Added April 7, 2021
See their solution, V47n03, pages 159-160 (they give two solutions)
The first solution starts from identity
\frac{1}{1-t}=1+t^2+t+\frac{t^3}{1-t}
and for the interval t\in ( 0,\;1 ) we have
t+\frac{t^3}{1-t}\geq 3t^2\;(\Leftrightarrow t(2t-1)^2 \geq 0).
N.CiP In fact, it is equivalent to inequality (1).
The second solution uses that for x \in (0, \; 64) we have
\frac{1}{8-\sqrt{x}} \geq \frac {x+16}{128}\; \Leftrightarrow \sqrt{x}(\sqrt{x}-4)^2 \geq 0.
N. CiP It is essentially the inequality (2).
= end added=
Niciun comentariu:
Trimiteți un comentariu