Triangle Inequality: (Bangladesh Online Math Camp)

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TOWFIQUL
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Re: Triangle Inequality: (Bangladesh Online Math Camp)

Unread post by TOWFIQUL » Tue Oct 25, 2011 10:31 pm

Anybody solved the exercise 1.10?

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Labib
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Re: Triangle Inequality: (Bangladesh Online Math Camp)

Unread post by Labib » Tue Oct 25, 2011 10:33 pm

I was explaining your third case. So $y \geq 0$
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Re: Triangle Inequality: (Bangladesh Online Math Camp)

Unread post by iPavel » Tue Oct 25, 2011 10:47 pm

আমি মোবাইল থেকে ব্যাবহার করি তাই latex ব্যাবহার করতে পারিনা।

(উল্লেখ্য আপনার latex ব্যাবহার করে যে ইকুয়েশন গুলো লেখেন সেগুলো আমার মোবাইলে বিদঘুটে কিছু চিহ্ন আকারে দেখা যায়।এর কোন প্রতিকার আছে?)

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Re: Triangle Inequality: (Bangladesh Online Math Camp)

Unread post by iPavel » Tue Oct 25, 2011 10:58 pm

@toweikul
assume without loss of genarility,x greater than or equal y then (x)^1/2 is greater than or equal (y)^1/2

now consider two set,A={x,y} and B={(x)^1/2,(y)^1/2}

using rearrangement inequality,x(x)^1/2+y(y)^1/2 is greater than or equal x(y)^1/2+y(x)^1/2=(x)^1/2*(y)^1/2((x)^1/2+(y)^1/2)

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Re: Triangle Inequality: (Bangladesh Online Math Camp)

Unread post by sampad » Wed Oct 26, 2011 12:58 am

This inequality is taught in class eleven in real numbers. But no teacher ever told us that it is called The Triangle inequality!
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Re: Triangle Inequality: (Bangladesh Online Math Camp)

Unread post by *Mahi* » Wed Oct 26, 2011 10:35 am

It is called the triangle inequality because it can also be represented as the form of the classic geometric inequality ,
for $\triangle ABC$, sum of any two sides is greater than the third.
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Re: Triangle Inequality: (Bangladesh Online Math Camp)

Unread post by mugdho.snigdho » Wed Oct 26, 2011 11:14 am

Could u plz state the problem 1.10 (i'm in the class & apart from the book right now ) ?

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Re: Triangle Inequality: (Bangladesh Online Math Camp)

Unread post by nafistiham » Wed Oct 26, 2011 12:02 pm

unfortunately, i couldn't solve this problem $1.10$ .again stating the problem for mugdho.snigdho and myself.hoping to get some hint

if $x,y> 0$, then prove that $\sqrt{\frac{x^{2}}{y}} + \sqrt{\frac{y^{2}}{x}}\geqslant \sqrt{x}+\sqrt{y}$
\[\sum_{k=0}^{n-1}e^{\frac{2 \pi i k}{n}}=0\]
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Labib
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Re: Triangle Inequality: (Bangladesh Online Math Camp)

Unread post by Labib » Wed Oct 26, 2011 12:24 pm

Tiham,
Would you like some hints?
Here are some if you want...
(a) Try squaring the sides and then simplify...
It should come to this-
It is enough to proof that,
$x^3+y^3\geq xy(x+y)$

(b) Now, without loss of generality, assume $x\geq y$,

(c) Now, substituting $x$ with $y+k$ follows the result.
The equality holds when $k=0$.
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Re: Triangle Inequality: (Bangladesh Online Math Camp)

Unread post by Hasib » Wed Oct 26, 2011 12:29 pm

To solve the proble 1.10, you have to define two case while $x \geq y$ and $x<y$ and u can try by squaring two side. :D
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