Jump to content
Sign in to follow this  
KTO Matematika

Kontes Terbuka Olimpiade Matematika Juli 2016 - Bagian B Nomor 3

Recommended Posts

On 7/27/2016 at 1:26 PM, lawmv7 said:
  Reveal hidden contents

 

Oyaa? :o boleh bagi caranya kk? hehe :)

Share this post


Link to post
Share on other sites

Cara Cauchy Schwarz :

 

Spoiler

$\begin{align*}n^2=(1+1+...+1)^2&\leq \left(1+x+x^2+...+x^{n-1}\right)\left(1+1/x+1/x^2+...+1/x^{n-1}\right)\\ &\leq \frac{(x^n-1)(1-1/x^n)}{(x-1)(1-1/x)} \\ &= \frac{x^n+\frac{1}{x^n}-2}{x+\frac{1}{x}-2}\end{align*}$

 

 

Cara AM-GM 

Spoiler

$\frac{x^n-1}{x-1}=1+x+...+x^{n-1}\geq n x^{n(n-1)/2n}$

Dikuadratin diperoleh 

\[n^2x^{n-1} \leq \frac{(x^n-1)^2}{(x-1)^2}\]

Diperoleh

 

\[n^2 \leq \frac{\frac{\left(x^n-1\right)^2 }{x^n}}{\frac{(x-1)^2}{x}}=\frac{x^n+\frac{1}{x^n}-2}{x+\frac{1}{x}-2}\]

 

 

 

Btw keduanya dengan asumsi $x\neq 1$ .. (otherwise diperoleh kesamaan)

Edited by Adri

Share this post


Link to post
Share on other sites

Itu ada yg salah ketik kayaknya pak, yg cara AM-GM penyebutnya harusnya (x-1)^2/ x = ...

On 29/7/2016 at 7:34 PM, Adri said:

Cara Cauchy Schwarz :

 

  Reveal hidden contents

n2=(1+1+...+1)2(1+x+x2+...+xn1)(1+1/x+1/x2+...+1/xn1)(xn1)(11/xn)(x1)(11/x)=xn+1xn2x+1x2n2=(1+1+...+1)2≤(1+x+x2+...+xn−1)(1+1/x+1/x2+...+1/xn−1)≤(xn−1)(1−1/xn)(x−1)(1−1/x)=xn+1xn−2x+1x−2

 

 

Cara AM-GM 

  Hide contents

xn1x1=1+x+...+xn1nxn(n1)/2nxn−1x−1=1+x+...+xn−1≥nxn(n−1)/2n

Dikuadratin diperoleh 

 

n2xn1(xn1)2(x1)2n2xn−1≤(xn−1)2(x−1)2

 

Diperoleh

 

 

n2(xn1)2xn(x1)2x2=xn+1xn2x+1x2n2≤(xn−1)2xn(x−1)2x2=xn+1xn−2x+1x−2

 

 

 

 

Btw keduanya dengan asumsi x1x≠1 .. (otherwise diperoleh kesamaan)

 

Share this post


Link to post
Share on other sites
1 hour ago, Syukri Lukman said:

Itu ada yg salah ketik kayaknya pak, yg cara AM-GM penyebutnya harusnya (x-1)^2/ x = ...

 

ohh iya, terima kasih :)

Share this post


Link to post
Share on other sites

Create an account or sign in to comment

You need to be a member in order to leave a comment

Create an account

Sign up for a new account in our community. It's easy!

Register a new account

Sign in

Already have an account? Sign in here.

Sign In Now

Sign in to follow this  

×