Barisal Secondary 2013 / 8

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Labib
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Barisal Secondary 2013 / 8

Unread post by Labib » Mon Jan 13, 2014 7:42 pm

The incenter of triangle $ABC$ is $I$ and inradius is $2$. What is the smallest possible
value of $AI+BI+CI$ ?
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Re: Barisal Secondary 2013 / 8

Unread post by photon » Tue Jan 14, 2014 2:25 pm

I solved it before , so giving a hint -
express $AI,BI,CI$ with angles and sides and then use AM-GM inequality.
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*Mahi*
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Re: Barisal Secondary 2013 / 8

Unread post by *Mahi* » Tue Jan 14, 2014 2:31 pm

Or Jensen, if you want :)
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Re: Barisal Secondary 2013 / 8

Unread post by bristy1588 » Tue Jan 14, 2014 10:04 pm

Mahi, Why don't you explain the Jensen's Inequality ?
There is a formula that I think you can find in Plane Euclidean Geometry.
$ AI = \frac{r}{sin(A/2) } $

$ BI = \frac{r}{sin(B/2)} $

$ CI = \frac{r}{sin(C/2)} $

Here $r$ is the inradius. In the interval $[0, \pi]$, $ sin(\frac{x}{2}) $ is a concave function.
Applying Jensen's inequality, $ x = 30 $ will give the solution.

For more information on Jensons http://www.artofproblemsolving.com/Wiki ... Inequality
you can check out this link.
Last edited by bristy1588 on Wed Jan 15, 2014 11:39 am, edited 1 time in total.
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Re: Barisal Secondary 2013 / 8

Unread post by Fatin Farhan » Wed Jan 15, 2014 10:22 am

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Re: Barisal Secondary 2013 / 8

Unread post by *Mahi* » Wed Jan 15, 2014 11:03 am

bristy1588 wrote: Here $r$ is the inradius. In the interval $[0, \pi]$, $ sin(\frac{x}{2}) $ is a convex function.
Applying Jensen's inequality, $ x = 30 $ will give the solution.
In the interval $[0, \frac \pi 2]$, $ \text{cosec } x$ is a convex function (and $\sin \frac x 2$ is concave in $[0,\pi]$).

@Fatin:
So, following the link on Jensen's inequality, $\dfrac {\text{cosec } \frac A 2 + \text{cosec } \frac B 2 + \text{cosec } \frac C 2} 3 \geq \text{cosec } \frac {\frac A 2 + \frac B 2 + \frac C 2} 3 = \text{cosec } 30$, and the rest is straightforward.
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Re: Barisal Secondary 2013 / 8

Unread post by asif e elahi » Wed Jan 15, 2014 7:38 pm

What is Jensen's inequality?What is convex and concave function?

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Re: Barisal Secondary 2013 / 8

Unread post by Mursalin » Wed Jan 15, 2014 9:42 pm

I worked on this one. And here I've posted a solution that doesn't resort to Jensen's inequality.
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Re: Barisal Secondary 2013 / 8

Unread post by Phlembac Adib Hasan » Thu Jan 16, 2014 12:56 pm

@asif কোন একটা ফাংশনের গ্রাফে দুইটা বিন্দু নিবা। বিন্দু দুইটা যোগ করে একটা সরলরেখা আকবা। এখন এই দুই বিন্দুর মাঝে ফাংশনের গ্রাফ সবসময় যদি ওই সরলরেখার উপরে থাকে তাহলে ফাংশনটা ওই দুই বিন্দুর মাঝে কনকেভ, আর যদি নিচে থাকে তাহলে ওই দুই বিন্দুর মাঝে কনভেক্স।
উদাহরণ-
Image

"a twice-differentiable function f, if the second derivative, f ′′(x), is positive (or, if the acceleration is positive), then the graph is convex; if f ′′(x) is negative, then the graph is concave"
Last edited by Phlembac Adib Hasan on Sun Jan 19, 2014 11:30 am, edited 1 time in total.
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Re: Barisal Secondary 2013 / 8

Unread post by asif e elahi » Thu Jan 16, 2014 1:48 pm

Phlembac Adib Hasan wrote:@asif কোন একটা ফাংশনের গ্রাফে দুইটা বিন্দু নিবা। বিন্দু দুইটা যোগ করে একটা সরলরেখা আকবা। এখন এই দুই বিন্দুর মাঝে ফাংশনের গ্রাফ সবসময় যদি ওই সরলরেখার উপরে থাকে তাহলে ফাংশনটা কনকেভ, আর যদি নিচে থাকে তাহলে কনভেক্স।
উদাহরণ- http://upload.wikimedia.org/wikipedia/e ... aveDef.png

"a twice-differentiable function f, if the second derivative, f ′′(x), is positive (or, if the acceleration is positive), then the graph is convex; if f ′′(x) is negative, then the graph is concave"
যদি ফাংশনের গ্রাফ ঢেউ এর মতো হয় মানে একবার সরলরেখার উপরে আরেকবার নিচে দিয়া যায় ৷

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