BdMO National Junior 2011/5
Problem 5:
Let, $A=211$ and $B=106^{211}$, which one is larger? Show logic.
($n!$ denotes the product of all the integers from $1$ to $n$. That means $n! =1\times 2 \times 3 \times 4 \times \cdots \times n$. For example $5!=1\times 2 \times 3 \times 4 \times 5 =120$.)
Let, $A=211$ and $B=106^{211}$, which one is larger? Show logic.
($n!$ denotes the product of all the integers from $1$ to $n$. That means $n! =1\times 2 \times 3 \times 4 \times \cdots \times n$. For example $5!=1\times 2 \times 3 \times 4 \times 5 =120$.)
"Inspiration is needed in geometry, just as much as in poetry." -- Aleksandr Pushkin
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Re: BdMO National Junior 2011/5
I think in the question u have missed the factorial sign..
Re: BdMO National Junior 2011/5
Yes he did...
And the simplest hint to solve it is...
And the simplest hint to solve it is...
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Re: BdMO National Junior 2011/5
I don't get you...
how will it work???....
how will it work???....
Re: BdMO National Junior 2011/5
Dipika...
$1\cdot 211=(106-105)(106+105)=106^2-105^2<106^2$
Do the same thing with others.
(Edited:: Courtesy AM Rafi)
$1\cdot 211=(106-105)(106+105)=106^2-105^2<106^2$
Do the same thing with others.
(Edited:: Courtesy AM Rafi)
Last edited by Labib on Sun Dec 25, 2011 10:51 pm, edited 1 time in total.
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"When you have eliminated the impossible, whatever remains, however improbable, must be the truth." - Sherlock Holmes
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"When you have eliminated the impossible, whatever remains, however improbable, must be the truth." - Sherlock Holmes
- Abdul Muntakim Rafi
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Re: BdMO National Junior 2011/5
Labib edit the typing mistake... should be $106^2-105^2<106^2$
Man himself is the master of his fate...
Re: BdMO National Junior 2011/5
বুঝিনা... ঠিকমত বুঝায়া দ্যান।
গণিত অলেম্পিয়াডে প্রাইজ পাওয়াটাই আসল না। প্রাইজ সবসময় পায়না এমন অনেকেও অনেক ভাল।
পরিচিতি
পরিচিতি
Re: BdMO National Junior 2011/5
আমাদের প্রমাণ করা দরকার
$1\times 2 \times 3 \times \cdots 105 \times 106 \times 107 \times \cdots \times 211 < 106 \times \cdots \times 106$
এখন 106 কাটাকাটি করলে থাকে \[1\times 2 \times 3 \times \cdots 105 \times 107 \times \cdots \times 211 < 106 \times \cdots \times 106\]
আমরা বামপাশ থেকে একটা ছোট আর একটা বড় নিয়ে 105 টা জোড়া নিব-- $1 \times 211, 2 \times 210$ ইত্যাদি (সব জোড়ার যোগফল 212.)
সুতরাং এটা প্রমাণ করলেই হবে যে $x \times (212-x) < 106^2 \iff x^2-2.106+106^2>0 \iff (x-106)^2>0$ যেটা সত্য।
সুতরাং এরকম ১০৫ টা জোড়া গুণ করলেই উপরের inequality পাওয়া যাবে।
$1\times 2 \times 3 \times \cdots 105 \times 106 \times 107 \times \cdots \times 211 < 106 \times \cdots \times 106$
এখন 106 কাটাকাটি করলে থাকে \[1\times 2 \times 3 \times \cdots 105 \times 107 \times \cdots \times 211 < 106 \times \cdots \times 106\]
আমরা বামপাশ থেকে একটা ছোট আর একটা বড় নিয়ে 105 টা জোড়া নিব-- $1 \times 211, 2 \times 210$ ইত্যাদি (সব জোড়ার যোগফল 212.)
সুতরাং এটা প্রমাণ করলেই হবে যে $x \times (212-x) < 106^2 \iff x^2-2.106+106^2>0 \iff (x-106)^2>0$ যেটা সত্য।
সুতরাং এরকম ১০৫ টা জোড়া গুণ করলেই উপরের inequality পাওয়া যাবে।
"Inspiration is needed in geometry, just as much as in poetry." -- Aleksandr Pushkin
Please install LaTeX fonts in your PC for better looking equations,
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Please install LaTeX fonts in your PC for better looking equations,
learn how to write equations, and don't forget to read Forum Guide and Rules.