Barisal Secondary 2013 / 8
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The incenter of triangle $ABC$ is $I$ and inradius is $2$. What is the smallest possible
value of $AI+BI+CI$ ?
value of $AI+BI+CI$ ?
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Re: Barisal Secondary 2013 / 8
I solved it before , so giving a hint -
Try not to become a man of success but rather to become a man of value.-Albert Einstein
Re: Barisal Secondary 2013 / 8
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Nur Muhammad Shafiullah | Mahi
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- bristy1588
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Re: Barisal Secondary 2013 / 8
Mahi, Why don't you explain the Jensen's Inequality ?
Last edited by bristy1588 on Wed Jan 15, 2014 11:39 am, edited 1 time in total.
Bristy Sikder
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Re: Barisal Secondary 2013 / 8
আপু এক্টু বুঝায় দেন। মাথায় ঢুকসে না।
"The box said 'Requires Windows XP or better'. So I installed L$$i$$nux...:p"
Re: Barisal Secondary 2013 / 8
In the interval $[0, \frac \pi 2]$, $ \text{cosec } x$ is a convex function (and $\sin \frac x 2$ is concave in $[0,\pi]$).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.
@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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- asif e elahi
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Re: Barisal Secondary 2013 / 8
What is Jensen's inequality?What is convex and concave function?
Re: Barisal Secondary 2013 / 8
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
@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-
"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"
উদাহরণ-
"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.
Reason: for further clarification
Reason: for further clarification
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- asif e elahi
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Re: Barisal Secondary 2013 / 8
যদি ফাংশনের গ্রাফ ঢেউ এর মতো হয় মানে একবার সরলরেখার উপরে আরেকবার নিচে দিয়া যায় ৷Phlembac 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"